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 Statistical Learning


Differentially Private Markov Chain Monte Carlo

Neural Information Processing Systems

Our algorithm is based on a decomposition of the Barker acceptance test that allows evaluating the Rényi DP privacy cost of the acceptreject choice.










Firstorderexpansionofconvexregularized estimators

Neural Information Processing Systems

Such first order expansion implies that the risk ofˆβ is asymptotically the same as the risk ofη which leads to a precise characterization of the MSE ofˆβ; this characterization takes aparticularly simple form for isotropic design. Such first order expansion also leads to inference results based onˆβ. We provide sufficient conditions for theexistence ofsuch first order expansion forthree regularizers: theLasso inits constrainedform,thelassoinitspenalizedform,andtheGroup-Lasso.Theresults apply to general loss functions under some conditions and those conditions are satisfied for the squared loss in linear regression and for the logistic loss in the logisticmodel.