We present a private learner for halfspaces over an arbitrary finite domain @math with sample complexity @math . The building block for this learner is a differentially private algorithm for locating an approximate center point of @math points -- a high dimensional generalization of the median function. Our construction establishes a relationship between these two problems that is reminiscent of the relation between the median and learning one-dimensional thresholds [ FOCS '15]. This relationship suggests that the problem of privately locating a center point may have further applications in the design of differentially private algorithms. We also provide a lower bound on the sample complexity for privately finding a point in the convex hull. For approximate differential privacy, we show a lower bound of @math , whereas for pure differential privacy @math .
Most related to our work is the work on private learning and its sample and time complexity by @cite_29 @cite_22 @cite_29 @cite_30 @cite_7 @cite_17 @cite_11 @cite_18 @cite_28 @cite_13 . As some of these works demonstrate efficiency gaps between private and non-private learning, alternative models have been explored including semi-supervised learning ( @cite_16 ), learning multiple concepts ( @cite_27 ), and prediction ( @cite_5 , @cite_21 ).
@cite_30 @cite_18 @cite_11 @cite_22 @cite_7 @cite_28 @cite_29 @cite_21 @cite_27 @cite_5 @cite_16 @cite_13 @cite_17
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In 2008, Kasiviswanathan el al. defined private learning as a combination of PAC learning and differential privacy [16]. Informally, a private learner is applied to a collection of labeled individual information and outputs a hypothesis while preserving the privacy of each individual. gave a generic construction of private learners for (finite) concept classes, with sample complexity logarithmic in the size of the concept class. This sample complexity is higher than what is needed for non-private learners, hence leaving open the possibility that the sample complexity of private learning may be sometimes significantly higher than that of non-private learning. We give a combinatorial characterization of the sample size sufficient and necessary to privately learn a class of concepts. This characterization is analogous to the well known characterization of the sample complexity of non-private learning in terms of the VC dimension of the concept class. We introduce the notion of probabilistic representation of a concept class, and our new complexity measure RepDim corresponds to the size of the smallest probabilistic representation of the concept class. We show that any private learning algorithm for a concept class C with sample complexity m implies RepDim(C) = O(m), and that there exists a private learning algorithm with sample complexity m = O(RepDim(C)). We further demonstrate that a similar characterization holds for the database size needed for privately computing a large class of optimization problems and also for the well studied problem of private data release. Learning is a task that generalizes many of the analyses that are applied to collections of data, in particular, to collections of sensitive individual information. Hence, it is natural to ask what can be learned while preserving individual privacy. (in SIAM J. Comput., 40(3):793---826, 2011) initiated such a discussion. They formalized the notion of private learning, as a combination of PAC learning and differential privacy, and investigated what concept classes can be learned privately. Somewhat surprisingly, they showed that for finite, discrete domains (ignoring time complexity), every PAC learning task could be performed privately with polynomially many labeled examples; in many natural cases this could even be done in polynomial time. While these results seem to equate non-private and private learning, there is still a significant gap: the sample complexity of (non-private) PAC learning is crisply characterized in terms of the VC-dimension of the concept class, whereas this relationship is lost in the constructions of private learners, which exhibit, generally, a higher sample complexity. Looking into this gap, we examine several private learning tasks and give tight bounds on their sample complexity. In particular, we show strong separations between sample complexities of proper and improper private learners (such separation does not exist for non-private learners), and between sample complexities of efficient and inefficient proper private learners. Our results show that VC-dimension is not the right measure for characterizing the sample complexity of proper private learning. We also examine the task of private data release (as initiated by in STOC, pp. 609---618, 2008), and give new lower bounds on the sample complexity. Our results show that the logarithmic dependence on size of the instance space is essential for private data release. A private learner is an algorithm that given a sample of labeled individual examples outputs a generalizing hypothesis while preserving the privacy of each individual. In 2008, (FOCS 2008) gave a generic construction of private learners, in which the sample complexity is (generally) higher than what is needed for non-private learners. This gap in the sample complexity was then further studied in several followup papers, showing that (at least in some cases) this gap is unavoidable. Moreover, those papers considered ways to overcome the gap, by relaxing either the privacy or the learning guarantees of the learner. We suggest an alternative approach, inspired by the (non-private) models of semi-supervised learning and active-learning, where the focus is on the sample complexity of labeled examples whereas unlabeled examples are of a significantly lower cost. We consider private semi-supervised learners that operate on a random sample, where only a (hopefully small) portion of this sample is labeled. The learners have no control over which of the sample elements are labeled. Our main result is that the labeled sample complexity of private learners is characterized by the VC dimension. We present two generic constructions of private semi-supervised learners. The first construction is of learners where the labeled sample complexity is proportional to the VC dimension of the concept class, however, the unlabeled sample complexity of the algorithm is as big as the representation length of domain elements. Our second construction presents a new technique for decreasing the labeled sample complexity of a given private learner, while roughly maintaining its unlabeled sample complexity. In addition, we show that in some settings the labeled sample complexity does not depend on the privacy parameters of the learner. A private learner is an algorithm that given a sample of labeled individual examples outputs a generalizing hypothesis while preserving the privacy of each individual. In 2008, (FOCS 2008) gave a generic construction of private learners, in which the sample complexity is (generally) higher than what is needed for non-private learners. This gap in the sample complexity was then further studied in several followup papers, showing that (at least in some cases) this gap is unavoidable. Moreover, those papers considered ways to overcome the gap, by relaxing either the privacy or the learning guarantees of the learner. We suggest an alternative approach, inspired by the (non-private) models of semi-supervised learning and active-learning, where the focus is on the sample complexity of labeled examples whereas unlabeled examples are of a significantly lower cost. We consider private semi-supervised learners that operate on a random sample, where only a (hopefully small) portion of this sample is labeled. The learners have no control over which of the sample elements are labeled. Our main result is that the labeled sample complexity of private learners is characterized by the VC dimension. We present two generic constructions of private semi-supervised learners. The first construction is of learners where the labeled sample complexity is proportional to the VC dimension of the concept class, however, the unlabeled sample complexity of the algorithm is as big as the representation length of domain elements. Our second construction presents a new technique for decreasing the labeled sample complexity of a given private learner, while roughly maintaining its unlabeled sample complexity. In addition, we show that in some settings the labeled sample complexity does not depend on the privacy parameters of the learner.