Differential privacy is concerned about the prediction quality while measuring the privacy impact on individuals whose information is contained in the data. We consider differentially private risk minimization problems with regularizers that induce structured sparsity. These regularizers are known to be convex but they are often non-differentiable. We analyze the standard differentially private algorithms, such as output perturbation, Frank-Wolfe and objective perturbation. Output perturbation is a differentially private algorithm that is known to perform well for minimizing risks that are strongly convex. Previous works have derived excess risk bounds that are independent of the dimensionality. In this paper, we assume a particular class of convex but non-smooth regularizers that induce structured sparsity and loss functions for generalized linear models. We also consider differentially private Frank-Wolfe algorithms to optimize the dual of the risk minimization problem. We derive excess risk bounds for both these algorithms. Both the bounds depend on the Gaussian width of the unit ball of the dual norm. We also show that objective perturbation of the risk minimization problems is equivalent to the output perturbation of a dual optimization problem. This is the first work that analyzes the dual optimization problems of risk minimization problems in the context of differential privacy.
The privacy parameters @math and @math need to be small to ensure stronger privacy. @cite_14 suggest that @math is a small constant and @math should be @math . They also showed that @math does not depend on @math to achieve a fixed level of privacy.
@cite_14
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This paper investigates the relation between three different notions of privacy: identifiability, differential privacy, and mutual-information privacy. Under a unified privacy-distortion framework, where the distortion is defined to be the expected Hamming distance between the input and output databases, we establish some fundamental connections between these three privacy notions. Given a maximum allowable distortion @math , we define the privacy-distortion functions @math , @math , and @math to be the smallest (most private best) identifiability level, differential privacy level, and mutual information between the input and the output, respectively. We characterize @math and @math , and prove that @math for @math within certain range, where @math is a constant determined by the prior distribution of the original database @math , and diminishes to zero when @math is uniformly distributed. Furthermore, we show that @math and @math can be achieved by the same mechanism for @math within certain range, i.e., there is a mechanism that simultaneously minimizes the identifiability level and achieves the best mutual-information privacy. Based on these two connections, we prove that this mutual-information optimal mechanism satisfies @math -differential privacy with @math . The results in this paper reveal some consistency between two worst case notions of privacy, namely, identifiability and differential privacy, and an average notion of privacy, mutual-information 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 (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). It is not uncommon in the data anonymization literature to oppose the "old" @math k -anonymity model to the "new" differential privacy model, which offers more robust privacy guarantees. Yet, it is often disregarded that the utility of the anonymized results provided by differential privacy is quite limited, due to the amount of noise that needs to be added to the output, or because utility can only be guaranteed for a restricted type of queries. This is in contrast with @math k -anonymity mechanisms, which make no assumptions on the uses of anonymized data while focusing on preserving data utility from a general perspective. In this paper, we show that a synergy between differential privacy and @math k -anonymity can be found: @math k -anonymity can help improving the utility of differentially private responses to arbitrary queries. We devote special attention to the utility improvement of differentially private published data sets. Specifically, we show that the amount of noise required to fulfill @math ? -differential privacy can be reduced if noise is added to a @math k -anonymous version of the data set, where @math k -anonymity is reached through a specially designed microaggregation of all attributes. As a result of noise reduction, the general analytical utility of the anonymized output is increased. The theoretical benefits of our proposal are illustrated in a practical setting with an empirical evaluation on three data sets. A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Privacy can be rigorously quantified using the framework of differential privacy , which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism @math -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user @math , no matter what its side information and preferences, derives as much utility from @math as from interacting with a differentially private mechanism @math that is optimally tailored to @math .