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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A diffuser layer is interposed between layers of moisture absorbent material situated within a cover sheet. The diffuser layer comprises one or more tissue layers, each having a plurality of parallel corrugations to disperse moisture by capillary action. A compression seal is situated in the cover sheet separating two pad parts. The compression seal comprises first and second crimps, spaced by a section of the cover sheet. The ends of the pad parts are also sealed by crimping. The absorbent material is wood pulp impregnated with a starch polymer or an acrylic-based polymer to increase the absorbency thereof. An agent may be included in the pulp to prevent the biological degradation of urine. Substances may also be included in the pulp for absorbing volatile nitrogenous compounds and odors. In addition, a moisture or chemical sensitive indicator strip may be situated on the cover sheet.