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We consider machine learning, particularly regression, usinglocally-differentially private datasets. The Wasserstein distance is used todefine an ambiguity set centered at the empirical distribution of the datasetcorrupted by local differential privacy noise. The ambiguity set is shown tocontain the probability distribution of unperturbed, clean data. The radius ofthe ambiguity set is a function of the privacy budget, spread of the data, andthe size of the problem. Hence, machine learning with locally-differentiallyprivate datasets can be rewritten as a distributionally-robust optimization.For general distributions, the distributionally-robust optimization problem canrelaxed as a regularized machine learning problem with the Lipschitz constantof the machine learning model as a regularizer. For linear and logisticregression, this regularizer is the dual norm of the model parameters. ForGaussian data, the distributionally-robust optimization problem can be solvedexactly to find an optimal regularizer. This approach results in an entirelynew regularizer for training linear regression models. Training with this novelregularizer can be posed as a semi-definite program. Finally, the performanceof the proposed distributionally-robust machine learning training isdemonstrated on practical datasets.

Distributionally-Robust Machine Learning Using Locally Differentially-Private Data