Introduction

This work studies the relationship between private PAC learning and online learning.

Differentially-Private Learning. Statistical analyses and computer algorithms play significant roles in the decisions which shape modern society. The collection and analysis of individuals' data drives computer programs which determine many critical outcomes, including the allocation of community resources, decisions to give loans, and school admissions.

While data-driven and automated approaches have obvious benefits in terms of efficiency, they also raise the possibility of unintended negative impacts, especially against marginalized groups. This possibility highlights the need for responsible algorithms that obey relevant ethical requirements (see e.g. [O'N16]).

Differential Privacy (DP) [DMNS06] plays a key role in this context. Its initial (and primary) purpose was to provide a formal framework for ensuring individuals' privacy in the statistical analysis of large datasets. But it has also found use in addressing other ethical issues such as algorithmic fairness (see, e.g. [DHP + 12, CGKM19]).

Online Learning. Online learning is a well-studied branch of machine learning which addresses algorithms making real-time predictions on sequentially arriving data. Such tasks arise in contexts including recommendation systems and advertisement placement. The literature on this subject is vast and includes several books, e.g. [CL06,SS12,Haz16].

Online Prediction, or Prediction with Expert Advice is a basic setting within online learning. Let H = {h : X → {±1}} be a class of predictors (also called experts) over a domain X. Consider an algorithm which observes examples (x 1 , y 1 ) . . . (x T , y T ) ∈ X × {±1} in a sequential manner. More specifically, in each time step t, the algorithm first observes the instance x t , then predicts a label ŷt ∈ {±1}, and finally learns whether its prediction was correct. The goal is to minimize the regret, namely the number of mistakes compared to the best expert in H:

In this context, a class H is said to be online learnable if for every T , there is an algorithm that achieves sublinear regret o(T ) against any sequence of T examples. The Littlestone dimension is a combinatorial parameter associated to the class H which characterizes its online learnability [Lit87,BPS09]: H is online learnable if and only if it has a finite Littlestone dimension d < ∞. Moreover, the best possible regret R(T ) for online learning of H satisfies Ω(

Furthermore, if it is known that if one of the experts never errs (a.k.a the realizable setting), then the optimal regret is exactly d. 1 (The regret is referred to by mistake-bound in this context.)

Stability. While at a first glance it may seem that online learning and differentially-private learning have little to do with one another, a recent line of work has revealed a tight connection between the two [AS17, ALMT17, BLM19, NRW19, JMNR19, GHM19].

At a high-level, this connection appears to boil down to the notion of stability, which plays a key role in both topics. On one hand, the definition of differential privacy is itself a form of stability; it requires robustness of the output distribution of an algorithm when its input undergoes small changes. On the other hand, stability also arises as a central motif in online learning paradigms such as Follow the Perturbed Leader [KV02,KV05] and Follow the Regularized Leader [AHR08, SSS07,Haz16].

In their monograph [DR14a], Dwork and Roth identified stability as a common factor of learning and differential privacy: "Differential privacy is enabled by stability and ensures stability. . . we observe a tantalizing moral equivalence between learnability, differential privacy, and stability." This insight has found formal manifestations in several works. For example, Abernethy et al. used DP inspired stability methodology to derive a unified framework for proving state of the art bounds in online learning [ALMT17]. In the opposite direction, Agarwal and Singh showed that certain standard stabilization techniques in online learning imply differential privacy [AS17].

Stability plays a key role in this work as well. The direction that any class with a finite Littlestone dimension can be privately learned hinges on the following form of stability: for η > 0 and n ∈ N, a learning algorithm A is (n, η)-globally stable 2 with respect to a distribution D over examples if there exists an hypothesis h whose frequency as an output is at least η. Namely,

Our argument follows by showing that every H can be learned by a globally-stable algorithm with parameters η = 2 -2 O(d) , n = 2 O(d) , where d is the Littlestone dimension of H. As a corollary, we get an equivalence between global stability and differential privacy (which can be viewed as a form of local stability). That is, the existence of a globally-stable learner for H is equivalent to the existence of a differentially-private learner for it (and both are equivalent to having a finite Littlestone dimension).

Littlestone Dimension and Thresholds. The converse direction -that every DP-learnable class has a finite Littlestone dimension -utilizes an intimate relationship between thresholds and the Littlestone dimension: a class H has a finite Littlestone dimension if and only if it does not embed thresholds as a subclass (for a formal statement, see Theorem 10); this follows from a seminal result in model theory by Shelah [She78]. As explained in the preliminaries (Section 3), Shelah's 1 More precisely, there is a deterministic algorithm that makes no more than d mistakes, and for every deterministic algorithm there is a (realizable) input sequence on which it makes at least d mistakes. For randomized algorithms a slightly weaker lower bound of d/2 holds with respect to the expected number of mistakes. 2 The word global highlights a difference with other forms of algorithmic stability. Indeed, previous forms of stability such as DP and uniform hypothesis stability [BE02] are local in the sense that they require output robustness subject to local changes in the input. However, the property required by global stability captures stability with respect to resampling the entire input.

theorem is usually stated in terms of orders and ranks. Chase and Freitag [CF18] noticedfoot_0 that the Littlestone dimension is the same as the model theoretic rank. Meanwhile, order translates naturally to thresholds. To make Theorem 10 more accessible for readers with less background in model theory, we provide a combinatorial proof in the appendix.

Littlestone Classes. It is natural to ask which classes have finite Littlestone dimension. First, note that every finite class H has Littlestone dimension d ≤ log|H|. There are also many natural and interesting infinite classes with finite Littlestone dimension. For example, let X = F n be an n-dimensional vector space over a field F and let H ⊆ {±1} X consist of all (indicators of) affine subspaces of dimension ≤ d. The Littlestone dimension of H is d. More generally, any class of hypotheses that can be described by polynomial equalities of bounded degree has a bounded Littlestone dimension. 4 This can be generalized even further to classes that are definable in stable theories. This (different, still) notion of stability is deep and well-explored in model theory. We refer the reader to [CF19], Section 5.1 for more examples of stable theories and the Littlestone classes they correspond to.

Organization. The rest of this manuscript is organized as follows. In Section 2 we formally state our main results and discuss some implications and other related and subsequent work. Then, in Section 4 we prove the direction that differentially private learnable classes have a finite Littlestone dimension, and in Section 5 we prove the converse direction, that every Littlestone class is differentially private PAC learnable. Finally, Section 6 concludes the paper with some suggestions for future work.

Results

We next present our main results that yield an equivalence between private PAC learning and online learning. We note that the derived equivalence is qualitative in the sense that the gap between the best known lower and upper bounds for learning a class H is incredibly large: the lower bound is proportional to log * (d), whereas the upper bound is doubly exponential in d, where d is the Littlestone dimension of H. Our upper bound has recently been reduced to Õ(d 6 ) in subsequent work [GGKM20].

The rest of this section is organized as follows: Sections 2.1, 2.2, and 2.3 are dedicated to the relationship between differentially-private learning, Littlestone dimension, and online learning, and in Section 2.4 we discuss an implication for private boosting. Throughout this section some standard technical terms are used. For definitions of these terms we refer the reader to Section 3.

Private Learning Implies Finite Littlestone Dimension

We begin by the following statement that resolves an open problem in [FX15] and [BNSV15]:

Theorem 1 (Thresholds are not privately learnable). Let X ⊆ R and let A be a ( 1 16 , 1 16 )-accurate learning algorithm for the class of thresholds over X with sample complexity n which satisfies (ε, δ)differential privacy with ε = 0.1 and δ

In particular, the class of thresholds over an infinite X can not be learned privately.

We note that an upper bound which scales with (log * |X|)

In particular any class that is privately learnable has a finite Littlestone dimension.

On the Proof of Theorem 1

A common approach of proving impossibility results in computer science (and in machine learning in particular) exploits a Minmax principle, whereby one specifies a fixed hard distribution over inputs, and establishes the desired impossibility result for any algorithm with respect to random inputs from that distribution. As an example, consider the "No-Free-Lunch Theorem" which establishes that the VC dimension lower bounds the sample complexity of PAC-learning a class H. Here, the hard distribution is picked to be uniform on a shattered set of size d = VC(H), and the argument follows by showing that every learning algorithm must observe Ω(d) examples. (See e.g. Theorem 5.1 in [SSBD14].) Such "Minmax" proofs establish a stronger assertion: they apply even to algorithms that "know" the input-distribution. For example, the No-Free-Lunch Theorem applies even to learning algorithms that are designed given the knowledge that the marginal distribution is uniform over some shattered set.

Interestingly, such an approach is bound to fail in proving Theorem 1. The reason is that if the marginal distribution D X is fixed, then one can pick an /2-coverfoot_2 , which we denote by C ε/2 , for the class thresholds over X of size |C ε/2 | = O(1/ ), and use the exponential mechanism [MT07] to DP-learn the finite class C ε/2 with sample complexity that scales with log|C ε/2 | = O(log(1/ε)). Since C ε/2 is an ε-cover for the class of thresholds, the obtained algorithm PAC learns the class of thresholds in a differentially private manner. To conclude, there is no single distribution which is "hard" for all DP algorithms that learn thresholds.

To overcome this difficulty one must come up with a method of assigning to any given algorithm A a "hard" distribution D = D A which is tailored to A and witnesses Theorem 1 with respect to A. The challenge is that A can be arbitrary; e.g. it may be improper. 6 We refer the reader to [NSY18, NY19, BMN + 18] for a line of work which explores in detail a similar "failure" of the Minmax principle in the context of PAC learning with low mutual information.

The "method" which we use to prove Theorem 1 exploits Ramsey theory. In a nutshell, Ramsey theory provides tools which allow to detect, for any learning algorithm, a "largish"' set X ⊆ X such that the behavior of A on input samples from X is highly regular. Then, the uniform distribution over X is the "hard" distribution which is used to derive Theorem 1.

We note that similar applications of Ramsey theory in computer science date back to the 80's [MSM85]. For more recent usages see e.g. [Bun16, CDFS19, CHK + 19].

Finally, we note that in the proper case, [BNSV15] demonstrated an ensemble, namely a distribution over distributions, which is hard for every differentially private algorithm A: if one draws a random distribution D from the ensemble and runs A on an input-sample from D, then the expected error of A will be large. It is plausible that such a statement also holds for general (possibly improper) algorithm, and it would be interesting to find such a natural ensemble.

Finite Littlestone Dimension Implies Private Learning

The following statement provides an upper bound on the sample complexity of DP-learning H, which depends only on the Littlestone dimension of H and the privacy/utility parameters. In particular, it does not depend on |H|.

Theorem 3 (Littlestone Classes are Privately Learnable). Let H ⊆ {±1} X be a class with Littlestone dimension d, let ε, δ ∈ (0, 1) be privacy parameters, and let α, β ∈ (0, 1/2) be accuracy parameters. For

there exists an (ε, δ)-DP learning algorithm such that for every realizable distribution D, given an input sample S ∼ D n , the output hypothesis f = A(S) satisfies loss D (f ) ≤ α with probability at least 1 -β, where the probability is taken over S ∼ D n as well as the internal randomness of A.

A similar result holds in the agnostic setting:

Corollary 4 (Agnostic Learner for Littlestone Classes). Let H ⊆ {±1} X be a class with Littlestone dimension d, let ε, and δ ∈ (0, 1) be privacy parameters, and let α, β ∈ (0, 1/2) be accuracy parameters. For

there exists an (ε, δ)-DP learning algorithm such that for every distribution D, given an input sample S ∼ D n , the output hypothesis f = A(S) satisfies

with probability at least 1 -β, where the probability is taken over S ∼ D n as well as the internal randomness of A.

Corollary 4 follows from Theorem 3 by Theorem 2.3 in [ABMS20] which provides a general mechanism to transform a learner in the realizable setting to a learner in the agnostic settingfoot_4 . We note that formally the transformation in [ABMS20] is stated for a constant ε = O(1). Taking ε = O(1) is without loss of generality as a standard "secrecy-of-the-sample" argument can be used to convert this learner into one which is (ε, δ)-differentially private by increasing the sample size by a factor of roughly 1/ε and running the algorithm on a random subsample. See [KLN + 11, Vad17] for further details.

Online Learning Versus Differentially Private PAC Learning

Since Littlestone dimension characterizes online learnability [Lit87,BPS09], Theorem 2 and Theorem 3 imply an equivalence between differentially private PAC learning and online learning:

Theorem 5 (Private PAC Learning ≡ Online Prediction.). The following statements are equivalent for a class H ⊆ {±1} X :

1. H is online learnable.

2. H is approximate differentially-privately PAC learnable.

Theorem 5 directly follows from Theorem 2 (which gives 2 → 1) and Theorem 3 (which gives 1 → 2). We comment that a quantitative relation between the learning rates and mistake/regret bounds is also implied: for example, in the agnostic setting it is known that the optimal regret bound for H is Θd ( √ T ), where the Θd conceals a constant which depends on the Littlestone dimension of H [BPS09]. Similarly, we get that the optimal sample complexity of agnostically privately learning H is Θ d ( log(1/(βδ))

). We remark however that the above equivalence is mostly interesting from a theoretical perspective, and should not be regarded as an efficient transformation between online and private learning. Indeed, the Littlestone dimension dependencies concealed by the Θd (•) in the above bounds on the regret and sample complexities may be very different from one another. For example, there are classes for which the Θ d ( log(1/(βδ)) αε ) bound hides a poly(log * (d)) dependence, and the Θd ( √ T ) bound hides a Θ(d) dependence. One example which attains both of these dependencies is the class of thresholds over a linearly ordered domain of size 2 d [KLM + 19].

Global Stability

Our proof of Theorem 3 hinges on an intermediate property which we call global stability: Definition 6 (Global Stability). Let n ∈ N be a sample size and η > 0 be a global stability parameter. An algorithm A is (n, η)-globally-stable with respect to a distribution D if there exists an hypothesis h such that

While global stability is a rather strong property, it holds automatically for learning algorithms using a finite hypothesis class. By an averaging argument, every learner using n samples which produces a hypothesis in a finite hypothesis class H is (n, 1/|H|)-globally-stable. The following proposition generalizes "Occam's Razor" for finite hypothesis classes to show that global stability is enough to imply similar generalization bounds in the realizable setting.

Proposition 7 (Global Stability =⇒ Generalization). Let H ⊆ {±1} X be a class, and assume that A is a consistent learner for H (i.e. loss S (A(S)) = 0 for every realizable sample S). Let D be a realizable distribution such that A is (n, η)-globally-stable with respect to D, and let h be a hypothesis such that Pr S∼D n [A(S) = h] ≥ η, as guaranteed by the definition of global stability. Then,

Proof. Let α denote the loss of h, i.e. loss D (h) = α, and let E 1 denote the event that h is consistent with the input sample S. Thus,

By assumption, Pr[E 2 ] ≥ η. Now, since A is consistent we get that E 2 ⊆ E 1 , and hence that η ≤ (1 -α) n . This finishes the proof (using the fact that 1 -α ≤ e -α and taking the logarithm of both sides).

Another way to view global stability is in the context of pseudo-deterministic algorithms [GG11]. A pseudo-deterministic algorithm is a randomized algorithm which yields some fixed output with high probability. Thinking of a realizable distribution D as an instance on which PAC-learning algorithm has oracle access, a globally-stable learner is one which is "weakly" pseudo-deterministic in that it produces some fixed output with probability bounded away from zero. A different model of pseudo-deterministic learning, in the context of learning from membership queries, was defined and studied by Oliveira and Santhanam [OS18].

We prove Theorem 3 by constructing, for a given Littlestone class H, an algorithm A which is globally-stable with respect to every realizable distribution.

Boosting for Approximate Differential Privacy

Our characterization of private learnability in terms of the Littlestone dimension has new consequences for boosting the privacy and accuracy guarantees of differentially-private learners. Specifically, it shows that the existence of a learner with weak (but non-trivial) privacy and accuracy guarantees implies the existence of a learner with any desired privacy and accuracy parametersin particular, one with δ(n) = exp(-Ω(n)).

Theorem 8. There exists a constant c > 0 for which the following holds. Suppose that for some sample size n 0 there is an (ε 0 , δ 0 )-differentially private learner W for a class H satisfying the guarantee Pr

, and δ 0 ≤ c/n 2 0 log n 0 . Then there exists a constant C H such that for every α, β, ε, δ ∈ (0, 1) there exists an (ε, δ)differentially private learner for H with

Given a weak learner W as in the statement of Theorem 8, Theorem 2 imply that Ldim(H) is finite. Hence Theorem 3 allows us to construct a learner for H with arbitrarily small privacy and accuracy, yielding Theorem 8. The constant C H in the last line of the theorem statement suppresses a factor depending on Ldim(H).

Prior to our work, it was open whether arbitrary learning algorithms satisfying approximate differential privacy could be boosted in this strong a manner. We remark, however, that in the case of pure differential privacy, such boosting can be done algorithmically and efficiently. Specifically, given an (ε 0 , 0)-differentially private weak learner as in the statement of Theorem 8, one can first apply random sampling to improve the privacy guarantee to (pε 0 , 0)-differential privacy at the expense of increasing its sample complexity to roughly n 0 /p for any p ∈ (0, 1). The Boosting-for-People construction of Dwork, Rothblum, and Vadhan [DRV10] (see also [BCS20]) then produces a strong learner by making roughly T ≈ log(1/β)/α 2 calls to the weak learner. By composition of differential privacy, this gives an (ε, 0)-differentially private strong learner with sample complexity roughly

What goes wrong if we try to apply this argument using an (ε 0 , δ 0 )-differentially private weak learner? Random sampling still gives a (pε 0 , pδ 0 )-differentially private weak learner with sample complexity n 0 /p. However, this is not sufficient to improve the δ parameter of the learner as a function of the number of samples n. Thus the strong learner one obtains using Boostingfor-People still at best guarantees δ(n) = Õ(1/n 2 ). Meanwhile, Theorem 8 shows that the existence of a (0.1, Õ(1/n 2 ))-differentially private learner for a given class implies the existence of a (0.1, exp(-Ω(n))-differentially private learner for that class.

We leave it as an interesting open question to determine whether this kind of boosting for approximate differential privacy can be done algorithmically.

Related and Subsequent Work

In this work, we determine that the (approximately) differentially-privately learnable classes are exactly those which are online learnable. We note that PAC learnability under the much stronger constraint of pure differentially privacy has already been characterized by several natural parameters such as the probabilistic representation dimension [BNS19] and one-way communication complexity [FX15]. These characterizations even imply nearly tight bounds on the optimal sample complexity. This is in contrast with the equivalence derived in this work whose implied upper and lower bounds on the sample complexity are extremely far away from each other.

Subsequent to our work, Ghazi, Golowich, Kumar, and Manurangsi [GGKM20] gave a significantly improved upper bound of Õ(d 6 ) on the sample complexity of learning any class with Littlestone dimension d. Moreover, their learning algorithm is proper. There is still an enormous gap between this and our lower bound of Ω(log * d), but both the upper and lower bound are within polynomial factors of the best possible sample complexity bounds that depend only on the Littlestone dimension. Thus, despite the fact that DP learnability is characterized by the finiteness of the Littlestone dimension, it remains wide open to find meaningful quantitative bounds on the sample complexity of DP learning. This is discussed in more detail in Section 5.4, where we suggest directions for future work.

Subsequent work has also extended the connection between online learning, global stability, and private learning to settings beyond binary classification. The private learnability of Littlestone classes has been studied in multiclass classification [JKT20, BGS21], real-valued classification (regression) [JKT20,Gol21], quantum state learning [AQS21], and the online private learning model [GL21].

Ghazi, Kumar, and Manurangsi [GKM21] used a generalization of global stability to derive private learning algorithms for datasets where each individual contributes multiple samples. Global stability is also related to a definition of reproducibility for machine learning algorithms put forth by Impagliazzo, Lei, Pitassi, and Sorrell [ILPS22].

Finally, several papers have studied the question of whether computationally efficient reductions exist between online and private learning. Gonen, Hazan, and Moran [GHM19] gave an efficient compiler from low sample-complexity pure private learners to online learners, while Bun [Bun20] showed that under cryptographic assumptions, such a reduction cannot exist in general.

Preliminaries

PAC Learning

We use standard notation from statistical learning; see, e.g., [SSBD14]. Let X be any "domain" set and consider the "label" set Y = {±1}. A hypothesis is a function h : X → Y , which we alternatively write as an element of Y X . An example is a pair (x, y) ∈ X × Y . A sample S is a finite sequence of examples. We also use the following notation: for samples S, T , let S • T denote the combined sample obtained by appending T to the end of S.

Definition 9 (Population & Empirical Loss). Let D be a distribution over X × {±1}. The population loss of a hypothesis h : X → {±1} is defined by

Let S = (x i , y i ) n i=1 be a sample. The empirical loss of h with respect to S is defined by

Let H ⊆ Y X be a hypothesis class. A sample S is said to be realizable by H if there is h ∈ H such that loss S (h) = 0. A distribution D is said to be realizable by H if there is h ∈ H such that loss D (h) = 0. A learning algorithm A is a (possibly randomized) mapping taking input samples to output hypotheses. We denote by A(S) the distribution over hypotheses induced by the algorithm when the input sample is S. We say that A learnsfoot_5 a class H with α-error, (1 -β)confidence, and sample-complexity m if for every realizable distribution D:

For brevity if A is a learning algorithm with α-error and (1 -β)-confidence we will say that A is an (α, β)-accurate learner.

Online Learning

Littlestone Dimension. The Littlestone dimension is a combinatorial parameter that captures mistake and regret bounds in online learning [Lit87,BPS09].foot_6 Its definition uses the notion of mistake trees. A mistake tree is a binary decision tree whose internal nodes are labeled by elements of X. Any root-to-leaf path in a mistake tree can be described as a sequence of examples (x 1 , y 1 ), ..., (x d , y d ), where x i is the label of the i'th internal node in the path, and y i = +1 if the (i + 1)'th node in the path is the right child of the i'th node and y i = -1 otherwise. We say that a mistake tree T is shattered by H if for any root-to-leaf path (x 1 , y 1 ), ..., (x d , y d ) in T there is an h ∈ H such that h(x i ) = y i for all i ≤ d (see Figure 1). The Littlestone dimension of H, denoted Ldim(H), is the depth of largest complete tree that is shattered by H. We say that H is a Littlestone class if it has finite Littlestone dimension.

Littlestone Dimension and Thresholds. Recently, Chase and Freitag [CF18] noticed that the Littlestone dimension coincides with a model-theoretic measure of complexity, Shelah's 2-rank.

A classical theorem of Shelah connects bounds on 2-rank (Littlestone dimension) to bounds on the so-called order property in model theory. The order property corresponds naturally to the concept of thresholds. Let H ⊆ {±1} X be an hypothesis class. We say that H contains k thresholds if there are x 1 , . . . , x k ∈ X and h 1 , . . . , h k ∈ H such that h i (x j ) = 1 if and only if i ≤ j for all i, j ≤ k.

Shelah's result (part of the so-called Unstable Formula Theoremfoot_7 ) [She78, Hod97], which we use in the following translated form, provides a simple and elegant connection between Littlestone dimension and thresholds. For completeness, we provide a combinatorial proof of Theorem 10 in Appendix A.

In the context of model theory, Theorem 10 is used to establish an equivalence between finite Littlestone dimension and stable theories. It is interesting to note that an analogous connection between theories that are called NIP theories and VC dimension has also been previously observed and was pointed out by [Las92]; this in turn led to results in Learning theory: in particular within the context of compression schemes [LS13] but also some of the first polynomial bounds for the VC dimension for sigmoidal neural networks [KM97].

Mistake Bound and the Standard Optimal Algorithm (SOA). The simplest setting in which learnability is captured by the Littlestone dimension is called the mistake-bound model [Lit87]. Let H ⊆ {±1} X be a fixed hypothesis class known to the learner. The learning process takes place in a sequence of trials, where the order of events in each trial t is as follows:

(i) the learner receives an instance x t ∈ X, (ii) the learner responses with a prediction ŷt ∈ {±1}, and (iii) the learner is told whether or not the response was correct.

We assume that the examples given to the learner are realizable in the following sense: For the entire sequence of trials, there is a hypothesis h ∈ H such that y t = h(x t ) for every instance x t and correct response y t . An algorithm in this model learns H with mistake bound M if for every realizable sequence of examples presented to the learner, it makes a total of at most M incorrect predictions.

Littlestone showed that the minimum mistake bound achievable by any online learner is exactly Ldim(H) [Lit87]. Furthermore, he described an explicit algorithm, called the Standard Optimal Algorithm (SOA), which achieves this optimal mistake bound.

Standard Optimal Algorithm (SOA)

Extending the SOA to non-realizable sequences. Our globally-stable learner for Littlestone classes will make use of an optimal online learner in the mistake bound model. For concreteness, we pick the SOA (any other optimal algorithm will also work). It will be convenient to extend the SOA to sequences which are not necessarily realizable by a hypothesis in H. We will use the following simple extension of the SOA to non-realizable samples:

Definition 11 (Extending the SOA to non-realizable sequences). Consider a run of the SOA on examples (x 1 , y 1 ), . . . , (x m , y m ), and let h t denote the predictor used by the SOA after seeing the first t examples (i.e., h t is the rule used by the SOA to predict in the (t + 1)'st trial). Then, after observing both x t+1 , y t+1 do the following:

• If the sequence (x 1 , y 1 ), . . . , (x t+1 , y t+1 ) is realizable by some h ∈ H then apply the usual update rule of the SOA to obtain h t+1 .

• Else, set h t+1 as follows: h t+1 (x t+1 ) = y t+1 , and h t+1 (x) = h t (x) for every x = x t+1 .

Thus, upon observing a non-realizable sequence, this update rule locally updates the maintained predictor h t to agree with the last example.

Differential Privacy

We use standard definitions and notation from the differential privacy literature. For more background see, e.g., the surveys [DR14a,Vad17]. For a, b, ε, δ

We say that two probability distributions p, q are (ε, δ)-indistinguishable if p(E) ≈ ε,δ q(E) for every event E.

Definition 12 (Private Learning Algorithm). A randomized algorithm

A : (X × {±1}) m → {±1} X is (ε, δ)-differentially-private if for every two samples S, S ∈ (X × {±1}) n that disagree on a single example, the output distributions A(S) and A(S ) are (ε, δ)-indistinguishable.

We emphasize that (ε, δ)-indistinguishability must hold for every such pair of samples, even if they are not generated according to a (realizable) distribution.

The parameters ε, δ are usually treated as follows: ε is a small constant (say 0.1), and δ is negligible, δ = n -ω(1) , where n is the input sample size. The case of δ = 0 is also referred to as pure differential privacy. Thus, a class H is privately learnable if it is PAC learnable by an algorithm A that is (ε(n), δ(n))-differentially private with ε(n) ≤ 0.1, and δ(n) ≤ n -ω(1) .

We will use the following corollary of the Basic Composition Theorem from differential privacy (see, e.g. Theorem 3.16 in [DR14b]).

Lemma 13 (Composition). [DKM + 06, DL09] If p, q are (ε, δ)-indistinguishable then for all k ∈ N, p k and q k are (kε, kδ)-indistinguishable, where p k , q k are the k-fold products of p, q (i.e. corresponding to k independent samples).

Private Empirical Learners. For the proof of Theorem 1 it will be convenient to consider the following task of minimizing the empirical loss.

Definition 14 (Empirical Learner). Algorithm A is (α, β)-accurate empirical learner for a hypothesis class H with sample complexity m if for every h ∈ H and for every sample S = ((x 1 , h(x 1 ), . . . , (x m , h(x m ))) ∈ (X × {±1}) m the algorithm A outputs a function f satisfying Pr

This task is simpler to handle than PAC learning, which is a distributional loss minimization task. Replacing PAC learning by this task does not lose generality; this is implied by the following result by [BNSV15].

Lemma 15. [[BNSV15], Lemma 5.9] Suppose ε < 1 and A is an ( , δ)-differentially private (α, β)accurate learning algorithm for a hypothesis class H with sample complexity m. Then there exists an ( , δ)-differentially private (α, β)-accurate empirical learner for H with sample complexity 9m.

Additional Notation

A sample S of an even length is called balanced if half of its labels are +1's and half are -1's.

For a sample S, let S X denote the underlying set of unlabeled examples: S X = x|(∃y) : (x, y) ∈ S . Let A be a randomized learning algorithm. It will be convenient to associate with A and S the function A S : X → [0, 1] defined by

Intuitively, this function represents the average hypothesis outputted by A when the input sample is S.

For the next definitions assume that the domain X is linearly ordered. Let S = ((x i , y i )) m i=1 be a sample. We say that S is increasing if x 1 < x 2 < . . . < x m . For x ∈ X define ord S (x) by |{i|x i ≤ x}|. Note that the set of points x ∈ X with the same ord S (x) form an interval whose endpoints are two consecutive examples in S (consecutive with respect to the order on X, i.e. there is no example x i between them). The tower function twr k (x) is defined by the recursion

The iterated logarithm, log (k) (x) is defined by the recursion

The function log * x equals the number of times the iterated logarithm must be applied before the result is less than or equal to 1. It is defined by the recursion

Private Learning Implies Finite Littlestone Dimension

In this section we prove that every class H which can be PAC-learned by a DP algorithm has a finite Littlestone dimension. This is achieved by establishing a lower bound on the sample complexity of privately learning H which depends on its Littlestone dimension (Theorem 2). The crux of this lower bound lies in Theorem 1, which provides a lower bound for the task of privately learning 1dimensional thresholds. This section is organized as follows. In Section 4.1 we provide an overview of the proof. Then, in Sections 4.2 and 4.3 we prove Theorems 1 and 2.

Proof overview

The starting point of the proof is Theorem 10, which asserts that if H has Littlestone dimension d, then it contains, as a subclass, at least some log d thresholds. In other words, the class of thresholds is "complete" in the sense that a lower bound on the sample complexity of DP learning thresholds yields a lower bound for classes with large Littlestone dimension.

Thus, consider an arbitrary differentially private algorithm A that learns the class of thresholds over an ordered domain X of size n. Our goal is to show a lower bound of Ω(log * n) on the sample complexity of A. A central challenge in the proof emerges because A may be improper and output arbitrary hypotheses (this is in contrast with proving impossibility results for proper algorithms where the structure of the learned class can be exploited).

The proof consists of two parts: (i) the first part handles the above challenge by showing that for any algorithm (in fact, for any mapping that takes input samples to output hypotheses) there is a large subset of the domain that is homogeneous with respect to the algorithm. This notion of homogeneity places useful restrictions on the algorithm on input samples from the homogeneous set. (ii) The second part of the argument utilizes the homogeneity of X ⊆ X to derive a lower bound on the sample complexity of the algorithm in terms of |X |.

We note that the Ramsey argument in the first part is quite general: it does not use the definition of differential privacy and could perhaps be useful in other sample complexity lower bounds. It is also worth noting that a Ramsey-based argument was used by [Bun16] in a weaker lower bound for DP learning thresholds in the proper case. In contrast to the first part, the second (and more technical) part of the proof is tailored specifically to the definition of differential privacy. We next outline each of these two parts.

Reduction to Homogeneous Sets. As discussed above, the first step in the proof is about identifying a large homogeneous subset of the input domain X on which we can control the output of A. To define homogeneity, recall from Section 3.4 that a sample S = ((x i , y i )) m i=1 of an even length is called balanced if half of its labels are +1's and half are -1's, and that S is said to be increasing if x 1 < x 2 < . . . < x m . Now, a subset X ⊆ X is called homogeneous with respect to A if there is a list of numbers p 0 , p 1 , . . . , p m such that for every increasing balanced sample S of points from X and for every x from X with ord S (x ) = i:

where γ is sufficiently small. For simplicity, in this proof overview we will assume that γ = 0. (In the proof γ is some O(1/m) -see Definition 16.) So, for example, if A is deterministic then h = A(S) is constant over each of the intervals defined by consecutive examples from S. See Figure 2 for an illustration.

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Figure 2: Depiction of two possible outputs of an algorithm over an homogeneous set, given two input samples from the set (marked in red). The number p i denote, for a given point x, the probability that h(x) = 1, where h ∼ A(S) is the hypothesis h outputted by the algorithm on input sample S. These probabilities depends (up to a small additive error) only on the interval that x belongs to. In the figure above we changed in the input the fourth example -this only affects the interval and not the values of the p i 's (again, up to a small additive error).

The derivation of a large homogeneous set follows by a standard application of Ramsey Theorem for hypergraphs using an appropriate coloring (Lemma 17).

Lower Bound for Homogenous Algorithms. We next assume that X = {1, . . . , k} is a large homogeneous set with respect to A (with γ = 0). We will obtain a lower bound on the sample complexity of A, denoted by m, by constructing a family P of distributions such that: (i) on the one hand |P| ≤ 2 Õ(m 2 ) , and (ii) on the other hand |P| ≥ Ω(k). Combining these inequalities yields a lower bound on m in terms of |X | = k and concludes the proof.

The construction of P proceeds as follows and is depicted in Figure 3: let S be an increasing balanced sample of points from X . Using the fact that A learns thresholds it is shown that for some i 1 < i 2 we have that p i 1 ≤ 1/3 and p i 2 ≥ 2/3. Thus, by a simple averaging argument there is some i 1 ≤ i ≤ i 2 such that p i -p i-1 ≥ Ω(1/m).

The last step in the construction is done by picking an increasing sample S such that the interval (x i-1 , x i+1 ) has size n = Ω(k). For x ∈ (x i-1 , x i+1 ), let S x denote the sample obtained by replacing x i with x in S. By restricting the output hypothesis to the interval (x i-1 , x i+1 ) (which is of size n), each output distribution A(S x ) can be seen as a distribution over the cube {±1} n . Thus, the family of distributions P consists of all distributions P x = A(S x ) for x ∈ (x i-1 , x i+1 ). Since A is private, it follows that P has the following two properties:

• P x , P x ∈ P are (ε, δ)-indistinguishable for all x , x ∈ (x i-1 , x i+1 ), and

, then for all P x ∈ P

It remains to show that Ω(k) ≤ |P| ≤ 2 Õ(m 2 ) . The lower bound follows directly from the definition of P. The upper bound requires a more subtle argument: it exploits the composition property for differenital privacy (see Lemma 13) via a privacy-breaching "attack" which is based on binarysearch. This argument appears in Lemma 21, whose proof is self-contained.

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Figure 3: An illustration of the definition of the family P . Given an homogeneous set and two consecutive intervals where there is a gap of at least Ω(1/m) between p i and p i-1 (here i = 4). The distributions in P correspond to the different positions of the i'th example, which separates between the (i -1)'th and the i'th intervals.

A Lower Bound for Privately Learning Thresholds

Proof of Theorem 1

The proof uses the following definition of homogeneous sets. Recall the definitions of balanced sample and of an increasing sample. In particular that a sample S = ((x 1 , y 1 ), . . . , (x m , y m )) of an even size is realizable (by thresholds), balanced, and increasing if and only if x 1 < x 2 < . . . < x m and the first half of the y i 's are -1 and the second half are +1.

Definition 16 (m-homogeneous set). A set X ⊆ X is m-homogeneous with respect to a learning algorithm A if there are numbers p i ∈ [0, 1], for 0 ≤ i ≤ m such that for every increasing balanced realizable sample S ∈ X × {±1} m and for every x ∈ X \ S X :

where i = ord S (x). The list (p i ) m i=0 is called the probabilities-list of X with respect to A. Proof of Theorem 1. Let A be a (1/16, 1/16)-accurate learning algorithm that learns the class of thresholds over X with m examples and is (ε, δ)-differential private with ε = 0.1, δ = 1 10 3 m 2 log m . By Lemma 15 we may assume without loss of generality that A is an empirical learner with the same privacy and accuracy parameters and sample size that is at most 9 times larger.

Theorem 1 follows from the next two lemmas which we prove later:

Lemma 17 (Every algorithm has large homogeneous sets). Let A be a (possibly randomized) algorithm that is defined over input samples of size m over a domain X ⊆ R with |X| = n. Then, there is a set X ⊆ X that is m-homogeneous with respect to A of size

Lemma 17 allows us to focus on a large homogeneous set with respect to A. The next Lemma implies a lower bound in terms of the size of a homogeneous set. For simplicity and without loss of generality assume that the homogeneous set is {1, . . . , k}.

Lemma 18 (Large homogeneous sets imply lower bounds for private learning). Let A be an (0.1, δ)differentially private algorithm with sample complexity m and δ ≤ 1 10 3 m 2 log m . Let X = {1, . . . , k} be m-homogeneous with respect to A. Then, if A empirically learns the class of thresholds over X with (1/16, 1/16)-accuracy, then

With these lemmas in hand, Theorem 1 follows by a short calculation: indeed, Lemma 17 implies the existence of an homogeneous set X with respect to A of size k ≥ log (m) (n)/2 O(m log m) . We then restrict A to input samples from the set X , and by relabeling the elements of X assume that X = {1, . . . , k} . Lemma 18 then implies that k = 2 O(m 2 log 2 m) . Together we obtain that

for some constant c > 0. Applying the iterated logarithm t = log * (2 c•m 2 log m ) = log * (m) + O(1) times on the inequality yields that log (m+t) (n) = log (m+log * (m)+O( 1)) (n) ≤ 1, and therefore log * (n) ≤ log * (m) + m + O(1), which implies that m ≥ Ω(log * n) as required.

Proof of Lemma 17

We next prove that every learning algorithm has a large homogeneous set. We will use the following quantitative version of Ramsey Theorem due to [ER52] (see also the book [GRS90], or Theorem 10.1 in the survey by [MS17]):

Theorem 19. [ER52] Let s > t ≥ 2 and q be integers, and let N ≥ twr t (3sq log q). Then for every coloring of the subsets of size t of a universe of size N using q colors there is a homogeneous subsetfoot_8 of size s.

Proof of Lemma 17. Define a coloring on the (m + 1)-subsets of X as follows. Let D = {x 1 < x 2 < . . . < x m+1 } be an (m + 1)-subset of X. For each i ≤ m + 1 let D -i = D \ {x i }, and let S -i denote the balanced increasing sample on D -i . Set p i to be the fraction of the form t 10 2 m that is closest to A S -i (x i ) (in case of ties pick the smallest such fraction). The coloring assigned to A is the list (p 1 , p 2 , . . . , p m+1 ).

Thus, the total number of colors is (10 2 m + 1) (m+1) . By applying Theorem 19 with t := m + 1, q := (10 2 m + 1) (m+1) , and N := n there is a set X ⊆ X of size

such that all m + 1-subsets of X have the same color. One can verify that X is indeed mhomogeneous with respect to A.

Proof of Lemma 18

The lower bound is proven by using the algorithm A to construct a family of distributions P with certain properties, and use these properties to derive that Ω(k) ≤ P ≤ 2 O(m 2 log 2 m) , which implies the desired lower bound.

Lemma 20. Let A, X , m, k as in Lemma 18, and set n = k -m. Then there exists a family P = {P i : i ≤ n} of distributions over {±1} n with the following properties:

1. Every P i , P j ∈ P are (0.1, δ)-indistinguishable.

2. There exists r ∈ [0, 1] such that for all i, j ≤ n:

Lemma 21. Let P, n, m, r as in Lemma 20. Then n ≤ 2 10 3 m 2 log 2 m . By the above lemmas, k -m = |P| ≤ 2 10 3 m 2 log 2 m , which implies that k = 2 O(m 2 log 2 m) as required. Thus, it remains to prove these lemmas, which we do next.

For the proof of Lemma 20 we will need the following claim:

Claim 22. Let (p i ) m i=0 denote the probabilities-list of X with respect to A. Then for some 0 < i ≤ m: p i -p i-1 ≥ 1 4m Proof. The proof of this claim uses the assumption that A empirically learns thresholds. Let S be a balanced increasing realizable sample such that S X = {x 1 < . . . < x m } ⊆ X are evenly spaced points on K (so, S = (x i , y i ) m i=1 , where y i = -1 for i ≤ m/2 and y i = +1 for i > m 2 ). A is an (α = 1/16, β = 1/16)-empirical learner and therefore its expected empirical loss on S is at most (1 -β) • α + β • 1 ≤ α + β = 1/8, and so:

This implies that there is m/2 ≤ m 1 ≤ m such that A S (x m 1 ) ≥ 3/4. Next, by privacy if we consider S the sample where we replace x m 1 by x m 1 + 1 (with the same label), we have that

Note that ord S (x m 1 ) = m 1 -1, hence by homogeneity: p m 1 -1 ≥ 2 3 -1 10 2 m . Similarly we can show that for some 1 ≤ m 2 ≤ m 2 we have p m 2 -1 ≤ 1 3 + 1 10 2 m . This implies that for some m 2 -1 ≤ i ≤ m 1 -1:

Proof of Lemma 20. Let i be the index guaranteed by Claim 22 such that p i -p i-1 ≥ 1/4m. Pick an increasing realizable sample S ∈ X × {±1} m so that the interval J ⊆ X between x i-1 and

is of size k -m. For every x ∈ J let S x be the neighboring sample of S that is obtained by replacing x with x i . This yields family of neighboring samples S x : x ∈ (x i-1 , x i+1 ) such that

• every two output-distributions A(S x ), A(S x ) are (ε, δ)-indistinguishable (because A satisfies (ε, δ) differential privacy).

• Set r = p i+1 +p i

2

. Then for all x, x ∈ J:

The proof is concluded by restricting the output of A to J, and identifying J with [n] and each output-distributions A(S x ) with a distribution over {±1} n .

Proof of Lemma 21. Set T = 10 3 m 2 log 2 m -1, and D = 10 2 m 2 log T . We want to show that n ≤ 2 T +1 . Assume towards contradiction that n > 2 T +1 . Consider the family of distributions Q i = P D i for i = 1, . . . , n. By Lemma 13, each Q i , Q j are (0.1D, δD)-indistinguishable. We next define a set of mutually disjoint events E i for i ≤ 2 T that are measurable with respect to each of the Q i 's. For a sequence of vectors v = (v 1 , . . . , v D ) in {±1} n we let v ∈ {±1} n be the threshold vector defined by

Given a point in the support of any of the Q i 's, namely a sequence v = (v 1 , . . . , v D ) of D vectors in {±1} n define a mapping B according to the outcome of T steps of binary search on v as follows: probe the n 2 'th entry of v; if it is +1 then continue recursively with the first half of v. Else, continue recursively with the second half of v. Define the mapping B = B(v) to be the entry that was probed at the T 'th step. The events E j correspond to the 2 T different outcomes of B. These events are mutually disjoint by the assumption that n > 2 T +1 .

Notice that for any possible i in the image of B, applying the binary search on a sufficiently large i.i.d sample v from P i would yield B(v) = i with high probability. Quantitatively, a standard application of Chernoff inequality and a union bound imply that the event

We claim that for all j ≤ n, and i in the image of B:

Note that adding the auxiliary example (x, y) forces A to make exactly d mistakes on S i •(x, y). Now, if y ∼ {±1} satisfies y = c(x) then by the mistake-bound argument it holds that A(S i • (x, y)) = c. Therefore, since Pr y∼{±1} [c(x) = y] = 1/2, it follows that Pr

and we are done. Similar reasoning can be used by induction to handle the remaining cases (the next one would be that Pr[M = d -2] ≥ 1 2d , and so on). As the number of mistakes reduces, we need to guess more labels, to enforce mistakes on the algorithm. As we guess more labels the success rate reduces, nevertheless we never need to make more then 2 d such guesses. (Note that the random contests performed by the algorithm can naturally be presented using the internal nodes of a binary tree of depth ≤ d ). The proof we present in Section 5.2 is based on a similar idea of performing random contests, although the construction becomes more complex to handle other issues, such as generalization, which were not addressed here. For more details we refer the reader to the complete argument in Section 5.2.

Step 2: Globally-Stable Learning =⇒ Differentially-Private Learning

Given a globally-stable learner A for a concept class H, we can obtain a differentially-private learner using standard techniques in the literature on private learning and query release. If A is a (η, m)-globally stable learner with respect to a distribution D, we obtain a differentiallyprivate learner using roughly m/η samples from that distribution as follows. We first run A on k ≈ 1/η independent samples, non-privately producing a list of k hypotheses. We then apply a differentially-private "Stable Histograms" algorithm [KKMN09,BNS16b] to this list which allows us to privately publish a short list of hypotheses that appear with frequency Ω(η). Global stability of the learner A guarantees that with high probability, this list contains some hypothesis h with small population loss. We can then apply a generic differentially-private learner (based on the exponential mechanism) on a fresh set of examples to identify such an accurate hypothesis from the short list.

Globally-Stable Learning of Littlestone Classes

Theorem Statement

The following theorem states that any class H with a bounded Littlestone dimension can be learned by a globally-stable algorithm.

Theorem 23. Let H be a hypothesis class with Littlestone dimension d ≥ 1, let α > 0, and set

Then there exists a randomized algorithm G : (X × {±1}) m → {±1} X with the following properties.

Let D be a realizable distribution and let S ∼ D m be an input sample. Then there exists a hypothesis f such

The distributions D k

The Algorithm G is obtained by running the SOA on a sample drawn from a carefully tailored distribution. This distribution belongs to a family of distributions which we define next. Each of these distributions can be sampled from using black-box access to i.i.d. samples from D. Recall that for a pair of samples S, T , we denote by S • T the sample obtained by appending T to the end of S. Define a sequence of distributions D k for k ≥ 0 as follows:

Let n denote an "auxiliary sample" size (to be fixed later) and let D denote the target realizable distribution over examples. The distributions D k = D k (D, n) are defined by induction on k as follows:

1. D 0 : output the empty sample with probability 1.

3. Else, D k is defined recursively by the following process:

(iii) If f 0 = f 1 then go back to step (i).

(iv) Else, pick x ∈ {x : f 0 (x) = f 1 (x)} and sample y ∼ {±1} uniformly.

(v) If f 0 (x) = y then output S 0 • T 0 • (x, y) and else output S 1 • T 1 • (x, y) .

Please see Figure 4 for an illustration of sampling S ∼ D k for k = 3. We next observe some basic facts regarding these distributions. First, note that whenever D k is well-defined, the process in Item 3 terminates with probability 1.

Let k be such that D k is well-defined and consider a sample S drawn from D k . The size of S is |S| = k • (n + 1). Among these k • (n + 1) examples there are k • n examples drawn from D and k examples which are generated in Item 3(iv). We will refer to these k examples as tournament examples. Note that during the generation of S ∼ D k there are examples drawn from D which do not actually appear in S. In fact, the number of such examples may be unbounded, depending on how many times Items 3(i)-3(iii) were repeated. In Section 5.2.3 we will define a "Monte-Carlo" variant of D k in which the number of examples drawn from D is always bounded. This Monte-Carlo variant is what we actually use to define our globally-stable learning algorithm, but we introduce the simpler distributions D k to clarify our analysis.

The k tournament examples satisfy the following important properties.

Observation 24. Let k be such that D k is well-defined and consider running the SOA on the concatenated sample S • T , where S ∼ D k and T ∼ D n . Then (x,y) 1. Each tournament example forces a mistake on the SOA. Consequently, the number of mistakes made by the SOA when run on S • T is at least k.

2. SOA(S • T ) is consistent with T .

The first item follows directly from the definition of x in Item 3(iv) and the definition of S in Item 3(v). The second item clearly holds when S • T is realizable by H (because the SOA is consistent). For non-realizable S • T , Item 2 holds by our extension of the SOA in Definition 11.

The Existence of Frequent Hypotheses. The following lemma is the main step in establishing global stability.

Lemma 25. There exists k ≤ d and an hypothesis f : X → {±1} such that

Proof. Suppose for the sake of contradiction that this is not the case. In particular, this means that D d is well-defined and that for every f :

(2)

We show that this cannot be the case when f = c is the target concept (i.e., for c ∈ H which satisfies loss D (c) = 0). Towards this end, we first show that with probability 2 -2 d+2 over S ∼ D d we have that all d tournament examples are consistent with c: for k ≤ d let ρ k denote the probability that all k tournament examples in S ∼ D k are consistent with c. We claim that ρ k satisfies the recursion

). Indeed, consider the event E k that (i) in each of S 0 , S 1 ∼ D k-1 , all k -1 tournament examples are consistent with c, and (ii) that

. Further, since y ∈ {±1} is chosen uniformly at random and independently of S 0 and S 1 , we have that conditioned on E k , c(x) = y with probability 1/2. Taken together we have that

. Since ρ 0 = 1 we get the recursive relation

, and ρ 0 = 1.

Thus, it follows by induction that for k ≤ d, ρ k ≥ 4 • 2 -2 k+1 : the base case is verified readily, and the induction step is as follows:

Therefore, with probability 2 -2 d+2 we have that S • T is consistent with c (because all examples in S • T which are drawn from D are also consistent with c). Now, since each tournament example forces a mistake on the SOA (Observation 24), and since the SOA does not make more than d mistakes on realizable samples, it follows that if all tournament examples in S ∼ D d are consistent with c then SOA(S) = SOA(S • T ) = c. Thus,

, which contradicts Equation 2 and finishes the proof.

Generalization. The next lemma shows that only hypotheses f that generalize well satisfy the conclusion of Lemma 25 (note the similarity of this proof with the proof of Proposition 7):

Lemma 26 (Generalization). Let k be such that D k is well-defined. Then every f such that

Proof. Let f be a hypothesis such that Pr S∼D k ,T ∼D n [SOA(S•T ) = f ] ≥ 2 -2 d+2 and let α = loss D (h). We will argue that 2 -2 d+2 ≤ (1 -α) n .

(3) Define the events A, B as follows.

1. A is the event that SOA(S • T ) = f . By assumption, Pr[A] ≥ 2 -2 d+2 .

2. B is the event that f is consistent with T . Since |T | = n, we have that Pr[B] = (1 -α) n .

Note that A ⊆ B: Indeed, SOA(S • T ) is consistent with T by the second item of Observation 24. Thus, whenever SOA(S • T ) = f , it must be the case that f is consistent with T . Hence, Pr[A] ≤ Pr[B], which implies Inequality 3 and finishes the proof (using the fact that 1 -α ≤ 2 -α and taking logarithms on both sides).

The Algorithm G

A Monte-Carlo Variant of D k Consider the following first attempt of defining a globallystable learner G: (i) draw i ∈ {0 . . . d} uniformly at random, (ii) sample S ∼ D i , and (iii) output SOA(S • T ), where T ∼ D n . The idea is that with probability 1/(d + 1) the sampled i will be equal to a number k satisfying the conditions of Lemma 25, so the desired hypothesis f guaranteed by this lemma (which also has low population loss by Lemma 26) will be outputted with probability at least 2 -2 d /(d + 1).

The issue here is that sampling f ∼ D i may require an unbounded number of samples from the target distribution D (in fact, D i may even be undefined). (i) Draw S 0 , S 1 ∼ Dk-1 and T 0 , T 1 ∼ D n independently.

(ii) Let f 0 = SOA(S 0 • T 0 ), f 1 = SOA(S 1 • T 1 ).

(iii) If f 0 = f 1 then go back to step (i).

(iv) Else, pick x ∈ {x : f 0 (x) = f 1 (x)} and sample y ∼ {±1} uniformly.

(v) If f 0 (x) = y then output S 0 • T 0 • (x, y) and else output S 1 • T 1 • (x, y) .

Note that Dk is well-defined for every k, even for k such that D k is undefined (however, for such k's the probability of outputting "Fail" may be large).

It Proof. Note that E[M 0 ] = 0 as D 0 deterministically produces the empty sample. We first show that for all 0

and then conclude the desired inequality by induction.

To see why Inequality 4 holds, let the random variable R denote the number of times Item 3(i) was executed during the generation of S ∼ D i+1 . That is, R is the number of times a pair S 0 , S 1 ∼ D i and a pair T 0 , T 1 ∼ D n were drawn. Observe that R is distributed geometrically with success probability θ, where:

where the last inequality follows because i < k and hence D i is well-defined, which implies that Pr S,T SOA(S • T ) = h ≤ 2 -2 d+2 for all h. Now, the random variable M i+1 can be expressed as follows:

where

Indeed, the probability that R ≥ j is (1 -θ) j-1 and conditioned on R ≥ j, in the j'th execution of Item 3(i) two samples from D i are drawn and two samples from D n are drawn. Thus

where the last inequality is true because θ ≥ 1 -2 -2 d+2 ≥ 1/2. This gives Inequality 4. Next, using that E[M 0 ] = 0, a simple induction gives

and the lemma follows by taking i + 1 = k.

Proof of Theorem 23. Our globally-stable learning algorithm G is defined as follows.

Tools from Differential Privacy

We begin by stating a few standard tools from the differential privacy literature which underlie our construction of a learning algorithm. Let X be a data domain and let S ∈ X n . For an element x ∈ X, define freq S Then on input S ∼ D n , algorithm GenericLearner outputs, with probability at least 1-β, a hypothesis ĥ ∈ H such that loss D ( ĥ) ≤ 2α.

Our formulation of the guarantees of this algorithm differ slightly from those of [KLN + 11], so we give its standard proof for completeness.

Proof of Lemma 29. The algorithm GenericLearner(S) samples a hypothesis h ∈ H with probability proportional to exp(-εn loss S (h)/2). This algorithm can be seen as an instantiation of the Exponential Mechanism [MT07]; the fact that changing one sample changes the value of loss S (h) by at most 1 implies that GenericLearner is ε-differentially private.

We now argue that GenericLearner is an accurate learner. Let E denote the event that the sample S satisfies the following conditions:

1. For every h ∈ H such that loss D (h) > 2α, it also holds that loss S (h) > 5α/3, and 2. For the hypothesis h * ∈ H satisfying loss D (h * ) ≤ α, it also holds that loss S (h * ) ≤ 4α/3. We claim that Pr[E] ≥ 1 -β/2 as long as n ≥ O(log(|H|/β)/α). To see this, let h ∈ H be an arbitrary hypothesis with loss D (h) > 2α. By a multiplicative Chernoff boundfoot_9 we have loss S (h) > 7α/4 with probability at least 1 -β/(4|H|) as long as n ≥ O(log(|H|/β)/α). Taking a union bound over all h ∈ H shows that condition 1. holds with probability at least 1 -β/4. Similarly, a multiplicative Chernoff bound ensures that condition 2 holds with probability at least 1 -β/4, so E holds with probability at least 1 -β/2. Now we show that conditioned on E, the algorithm GenericLearner(S) indeed produces a hypothesis h with loss D ( ĥ) ≤ 2α. This follows the standard analysis of the accuracy guarantees of the Exponential Mechanism. Condition 2 of the definition of event E guarantees that loss S (h * ) ≤ 4α/3. This ensures that the normalization factor in the definition of the Exponential Mechanism is at least exp(-2εαn/3). Hence by a union bound, Pr[loss S ( ĥ) > 5α/3] ≤ |H| • exp(-5εαn/6) exp(-2εαn/3) = |H|e -εαn/6 .

Taking n ≥ O(log(|H|/β)/αε) ensures that this probability is at most β/2. Given that loss( ĥ) ≤ 5α/3, Condition 1 of the definition of event E ensures that loss D ( ĥ) ≤ 2α. Thus, for n sufficiently large as described, we have overall that loss D ( ĥ) ≤ 2α with probability at least 1 -β.

Construction of a Private Learner

We now describe how to combine the Stable Histograms algorithm with the Generic Private Learner to convert any globally-stable learning algorithm into a differentially-private one.

Theorem 30. Let H be a concept class over data domain X. Let G : (X × {±1}) m → {±1} X be a randomized algorithm such that, for D a realizable distribution and S ∼ D m , there exists a hypothesis h such that Pr[G(S) = h] ≥ η and loss D (h) ≤ α/2. Then for some

there exists an (ε, δ)-differentially private algorithm M : (X × {±1}) n → {±1} X which, given n i.i.d. samples from D, produces a hypothesis ĥ such that loss D ( ĥ) ≤ α with probability at least 1-β.

Theorem 30 is realized via the learning algorithm M described below. Here, the parameter k = Õ log(1/ηβδ) ηε is chosen so that Lemma 28 guarantees Algorithm Hist succeeds with the stated accuracy parameters. The parameter n = Õ log(1/ηβ) αε is chosen so that Lemma 29 guarantees that GenericLearner succeeds on a list L of size |L| ≤ 2/η with the given accuracy and confidence parameters.

Taking η = 1/(d + 1)2

Conclusion

We conclude this paper with a few suggestions for future work.

1. Sharper Quantitative Bounds. Our upper bound on the differentially-private sample complexity of a class H has a double exponential dependence on the Littlestone dimension Ldim(H), while the lower bound by [ALMM19] depends on log * (Ldim(H)). The work by [KLM + 19] shows that for thresholds, the lower bound is nearly tight (up to a polynomial factor). In a followup work to this paper, [GGKM20] improved the upper bound to poly(Ldim(H)) (roughly, with an exponent of 6). This is also tight up to polynomial factors for some classes, in particular, those with maximal Littlestone dimension equal to log |H|.

However the tower-of-exponents gap between the upper bound and the lower bound remains essentially the same (with 2 fewer levels). We thus pose the following question:

Can every class H be privately learned with sample complexity poly(VC(H), log * (Ldim(H)))?

2. Characterizing Private Query Release. Another fundamental problem in differentiallyprivate data analysis is the query release, or equivalently, data sanitization problem: Given a class H and a sensitive dataset S, output a synthetic dataset Ŝ such that h(S) ≈ h( Ŝ) for every h ∈ H. In earlier versions of this work, we asked whether finite Littlestone dimension characterizes when this task is possible. This was shown to be true by [BLM19] and [GGKM20]. ([BLM19] showed how to transform a proper private learner to a sanitizer, and [GGKM20] proved that every Littlestone class can be learned properly.) However, as with private classification, massive quantitative gaps between the known upper and lower bounds remain.

3. Oracle-Efficient Learning. Neel, Roth, and Wu [NRW19] recently began a systematic study of oracle-efficient learning algorithms: Differentially-private algorithms which are computationally efficient when given oracle access to their non-private counterparts. The main open question left by their work is whether every privately learnable concept class can be learned in an oracle-efficient manner. Our characterization shows that this is possible if and only if Littlestone classes admit oracle-efficient learners.

4. General Loss Functions. It is natural to explore whether the equivalence between online and private learning extends beyond binary classification (which corresponds to the 0-1 loss) to regression and other real-valued losses. These more general loss functions have been studied in subsequent work [JKT20, AQS21, BGS21, Gol21], though the problem of exactly characterizing private learnability in the regression setting remains open.

tree rooted at its left child and a subtree of height h -1 of the tree rooted at its right child. Note that if T is a labeled tree and it is shattered by the class H, then any subtree T of it with the same labeling of its internal vertices is shattered by the class H. With this definition we prove the following simple lemma.

Lemma 31. Let p, q be positive integers and let T be a labeled binary tree of height p + q -1 whose internal vertices are colored by two colors, red and blue. Then T contains either a subtree of height p in which all internal vertices are red (a red subtree), or a subtree of height q in which all vertices are blue (a blue subtree).

Proof: We apply induction on p + q. The result is trivial for p = q = 1 as the root of T is either red or blue. Assuming the assertion holds for p + q < p + q, let T be of height p + q -1. Without loss of generality assume the root of t is red. If p = 1 we are done, as the root together with a leaf in the subtree of its left child and one in the subtree of its right child form a red subtree of height p. If p > 1 then, by the induction hypothesis, the tree rooted at the left child of the root of T contains either a red subtree of height p -1 or a blue subtree of height q, and the same applies to the tree rooted at the right child of the root. If at least one of them contains a blue subtree as above we are done, otherwise, the two red subtrees together with the root provide the required red subtree.

We can now prove the first part of the theorem, showing that if the Littlestone dimension of H is at least 2 t+1 -1 then H contains t + 2 thresholds. We apply induction on t. If t = 0 we have a tree of height 1 shattered by H. Its root is labeled by some variable x 0 and as it is shattered there are two functions h 0 , h 1 ∈ H so that h 0 (x 0 ) = 1, h 1 (x 0 ) = 0, meaning that H contains two thresholds, as needed. Assuming the desired result holds for t -1 we prove it for t, t ≥ 1. Let T be a labeled binary tree of height 2 t+1 -1 shattered by H. Let h be an arbitrary member of H and define a two coloring of the internal vertices of T as follows. If an internal vertex is labeled by x and h(x) = 1 color it red, else color it blue. Since 2 t+1 -1 = 2 • 2 t -1, Lemma 31 with p = q = 2 t implies that T contains either a red or a blue subtree T of height 2 t . In the first case define h 0 = h and let X be the set of all variables x so that h(x) = 1. Let x 0 be the root of T and let T be the subtree of T rooted at the left child of T . Let H be the set of all h ∈ H so that h (x 0 ) = 0. Note that H shatters the tree T , and that the depth of T is 2 t -1. We can thus apply the induction hypothesis and get a set of t + 1 thresholds h 1 , h 2 , . . . , h t+1 ∈ H and variables x 1 , x 2 , . . . , x t ∈ X so that h i (x j ) = 1 iff j ≥ i. Adding h 0 and x 0 to these we get the desired t + 2 thresholds. Similarly, if T contains a blue subtree T , define h t+1 = h and let X be the set of all variables x so that h(x) = 0. In this case denote the root of T by x t and let T be the subtree of T rooted at the right child of T . Let H be the set of all h ∈ H so that h (x t ) = 1. As before, H shatters the tree T whose depth is 2 t -1. By the induction hypothesis we get t + 1 thresholds h 0 , h 1 , . . . , h t and variables x 0 , x 1 , . . . , x t-1 ∈ X so that h i (x j ) = 1 iff j ≥ i, and the desired result follows by appending to them h t+1 and x t . This completes the proof.