We consider machine learning, particularly regression, using locally-differentially private datasets. The Wasserstein distance is used to define an ambiguity set centered at the empirical distribution of the dataset corrupted by local differential privacy noise. The radius of the ambiguity set is selected based on privacy budget, spread of data, and size of the problem. Machine learning with private dataset is rewritten as a distributionally-robust optimization. For general distributions, the distributionally-robust optimization problem can be relaxed as a regularized machine learning problem with the Lipschitz constant of the machine learning model as a regularizer. For Gaussian data, the distributionally-robust optimization problem can be solved exactly to find an optimal regularizer. Training with this regularizer can be posed as a semi-definite program.

我们考虑使用本地差异私有数据集的机器学习，尤其是回归。Wasserstein 距离用于定义以被局部差分隐私噪声破坏的数据集的经验分布为中心的模糊集。模糊集的半径是根据隐私预算、数据的传播和问题的大小来选择的。使用私有数据集的机器学习被重写为分布稳健的优化。对于一般分布，分布稳健优化问题可以放松为正则化机器学习问题，将机器学习模型的 Lipschitz 常数作为正则化器。对于高斯数据，可以精确求解分布稳健优化问题以找到最佳正则化器。