Online learning has been in the spotlight from the machine learning society for a long time. To handle massive data in Big Data era, one single learner could never efficiently finish this heavy task. Hence, in this paper, we propose a novel distributed online learning algorithm to solve the problem. Comparing to typical centralized online learner, the distributed learners optimize their own learning parameters based on local data sources and timely communicate with neighbors. However, communication may lead to a privacy breach. Thus, we use differential privacy to preserve the privacy of learners, and study the influence of guaranteeing differential privacy on the utility of the distributed online learning algorithm. Furthermore, by using the results from Kakade and Tewari (2009), we use the regret bounds of online learning to achieve fast convergence rates for offline learning algorithms in distributed scenarios, which provides tighter utility performance than the existing state-of-the-art results. In simulation, we demonstrate that the differentially private offline learning algorithm has high variance, but we can use mini-batch to improve the performance. Finally, the simulations show that the analytical results of our proposed theorems are right and our private distributed online learning algorithm is a general framework.
As for the study of differential privacy, there has been much research effort being devoted to how differential privacy can be used in existing learning algorithms. For example, @cite_3 presented the output perturbation and objective perturbation ideas about differential privacy in empirical risk minimization (ERM) classification. They achieved a good utility for ERM algorithm while guaranteed @math -differential privacy. Rajkumar and Agarwal @cite_16 extended differentially private ERM classification @cite_3 to differentially private ERM multiparty classification. More importantly, they analyzed the sequential and parallel composability problems while the algorithm guaranteed @math -differential privacy. @cite_18 proposed more efficient algorithms and tighter error bounds for ERM classification on the basis of @cite_3 .
@cite_18 @cite_16 @cite_3
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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 [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... 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). Privacy-preserving machine learning algorithms are crucial for the increasingly common setting in which personal data, such as medical or financial records, are analyzed. We provide general techniques to produce privacy-preserving approximations of classifiers learned via (regularized) empirical risk minimization (ERM). These algorithms are private under the e-differential privacy definition due to (2006). First we apply the output perturbation ideas of (2006), to ERM classification. Then we propose a new method, objective perturbation, for privacy-preserving machine learning algorithm design. This method entails perturbing the objective function before optimizing over classifiers. If the loss and regularizer satisfy certain convexity and differentiability criteria, we prove theoretical results showing that our algorithms preserve privacy, and provide generalization bounds for linear and nonlinear kernels. We further present a privacy-preserving technique for tuning the parameters in general machine learning algorithms, thereby providing end-to-end privacy guarantees for the training process. We apply these results to produce privacy-preserving analogues of regularized logistic regression and support vector machines. We obtain encouraging results from evaluating their performance on real demographic and benchmark data sets. Our results show that both theoretically and empirically, objective perturbation is superior to the previous state-of-the-art, output perturbation, in managing the inherent tradeoff between privacy and learning performance. While machine learning has proven to be a powerful data-driven solution to many real-life problems, its use in sensitive domains has been limited due to privacy concerns. A popular approach known as differential privacy offers provable privacy guarantees, but it is often observed in practice that it could substantially hamper learning accuracy. In this paper we study the learnability (whether a problem can be learned by any algorithm) under Vapnik's general learning setting with differential privacy constraint, and reveal some intricate relationships between privacy, stability and learnability. In particular, we show that a problem is privately learnable if an only if there is a private algorithm that asymptotically minimizes the empirical risk (AERM). In contrast, for non-private learning AERM alone is not sufficient for learnability. This result suggests that when searching for private learning algorithms, we can restrict the search to algorithms that are AERM. In light of this, we propose a conceptual procedure that always finds a universally consistent algorithm whenever the problem is learnable under privacy constraint. We also propose a generic and practical algorithm and show that under very general conditions it privately learns a wide class of learning problems. Lastly, we extend some of the results to the more practical (e, δ)-differential privacy and establish the existence of a phase-transition on the class of problems that are approximately privately learnable with respect to how small δ needs to be.