We investigate the direct-sum problem in the context of differentially private PAC learning: What is the sample complexity of solving k learning tasks simultaneously under differential privacy, and how does this cost compare to that of solving k learning tasks without privacy? In our setting, an individual example consists of a domain element x labeled by k unknown concepts (c1,...,ck). The goal of a multi-learner is to output k hypotheses (h1,...,hk) that generalize the input examples. Without concern for privacy, the sample complexity needed to simultaneously learn @math concepts is essentially the same as needed for learning a single concept. Under differential privacy, the basic strategy of learning each hypothesis independently yields sample complexity that grows polynomially with k. For some concept classes, we give multi-learners that require fewer samples than the basic strategy. Unfortunately, however, we also give lower bounds showing that even for very simple concept classes, the sample cost of private multi-learning must grow polynomially in k.
Differential privacy was defined in @cite_1 and the relaxation to approximate differential privacy is from @cite_18 . Most related to our work is the work on private learning and its sample complexity @cite_38 @cite_3 @cite_4 @cite_0 @cite_42 @cite_26 @cite_24 @cite_8 @cite_17 @cite_39 and the early work on sanitization @cite_32 . That many natural'' learning tasks can be performed privately was shown in the early work of @cite_38 and @cite_3 . A characterization for the sample complexity of pure-private learners was given in @cite_26 , in terms of a new combinatorial measure -- the Representation Dimension , that is, given a class @math , the number of samples needed and sufficient for privately learning @math is @math . Building on @cite_26 , Feldman and Xiao @cite_8 showed an equivalence between the representation dimension of a concept @math and the randomized one-way communication complexity of the evaluation problem for concepts from @math . Using this equivalence they separated the sample complexity of pure-private learners from that of non-private ones.
@cite_38 @cite_18 @cite_26 @cite_4 @cite_8 @cite_42 @cite_1 @cite_32 @cite_3 @cite_39 @cite_0 @cite_24 @cite_17
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In this work we analyze the sample complexity of classification by differentially private algorithms. Differential privacy is a strong and well-studied notion of privacy introduced by (2006) that ensures that the output of an algorithm leaks little information about the data point provided by any of the participating individuals. Sample complexity of private PAC and agnostic learning was studied in a number of prior works starting with (, 2008) but a number of basic questions still remain open, most notably whether learning with privacy requires more samples than learning without privacy. We show that the sample complexity of learning with (pure) differential privacy can be arbitrarily higher than the sample complexity of learning without the privacy constraint or the sample complexity of learning with approximate differential privacy. Our second contribution and the main tool is an equivalence between the sample complexity of (pure) differentially private learning of a concept class @math (or @math ) and the randomized one-way communication complexity of the evaluation problem for concepts from @math . Using this equivalence we prove the following bounds: 1. @math , where @math is the Littlestone's (1987) dimension characterizing the number of mistakes in the online-mistake-bound learning model. Known bounds on @math then imply that @math can be much higher than the VC-dimension of @math . 2. For any @math , there exists a class @math such that @math but @math . 3. For any @math , there exists a class @math such that the sample complexity of (pure) @math -differentially private PAC learning is @math but the sample complexity of the relaxed @math -differentially private PAC learning is @math . This resolves an open problem of (2013b). We compare the sample complexity of private learning [ 2008] and sanitization [ 2008] under pure @math -differential privacy [ TCC 2006] and approximate @math -differential privacy [ Eurocrypt 2006]. We show that the sample complexity of these tasks under approximate differential privacy can be significantly lower than that under pure differential privacy. We define a family of optimization problems, which we call Quasi-Concave Promise Problems, that generalizes some of our considered tasks. We observe that a quasi-concave promise problem can be privately approximated using a solution to a smaller instance of a quasi-concave promise problem. This allows us to construct an efficient recursive algorithm solving such problems privately. Specifically, we construct private learners for point functions, threshold functions, and axis-aligned rectangles in high dimension. Similarly, we construct sanitizers for point functions and threshold functions. We also examine the sample complexity of label-private learners, a relaxation of private learning where the learner is required to only protect the privacy of the labels in the sample. We show that the VC dimension completely characterizes the sample complexity of such learners, that is, the sample complexity of learning with label privacy is equal (up to constants) to learning without privacy. In this work we analyze the sample complexity of classification by differentially private algorithms. Differential privacy is a strong and well-studied notion of privacy introduced by [Lecture Notes in Comput. Sci. 3876, Springer, New York, 2006, pp. 265--284] that ensures that the output of an algorithm leaks little information about the data point provided by any of the participating individuals. Sample complexity of private probably approximately correct (PAC) and agnostic learning was studied in a number of prior works starting with [SIAM J. Comput., 40 (2011), pp. 793--826]. However, a number of basic questions remain open [A. Beimel, S. P. Kasiviswanathan, and K. Nissim, Lecture Notes in Comput. Sci. 5978, Springer, New York, 2006, pp. 437--454; K. Chaudhuri and D. Hsu, Proceedings of Conference in Learning Theory, 2011, pp. 155--186; A. Beimel, K. Nissim, and U. Stemmer, Proceedings of the 4th Conference on Innovations in Theoretical Computer Science, 2013, pp. 9... We prove new upper and lower bounds on the sample complexity of (a#x03B5;, a#x03B4;) differentially private algorithms for releasing approximate answers to threshold functions. A threshold function cx over a totally ordered domain X evaluates to cx(y) = 1 if y a#x2264; x, and evaluates to 0 otherwise. We give the first nontrivial lower bound for releasing thresholds with (a#x03B5;, a#x03B4;) differential privacy, showing that the task is impossible over an infinite domain X, and moreover requires sample complexity n a#x2265; (log* |X|), which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with n a#x2264; 2(1+o(1)) log* |X| samples. This improves the previous best upper bound of 8(1+o(1)) log* |X| (, RANDOM'13). Our sample complexity upper and lower bounds also apply to the tasks of learning distributions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with (a#x03B5;, a#x03B4;) differential privacy and learning without privacy. For properly learning thresholds in 'dimensions, this lower bound extends to n a#x2265; (l a#x2219; log* |X|). To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database D of elements from X, the interior point problem asks for an element between the smallest and largest elements in D. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy.