Improved subsample-and-aggregate via the private modified winsorized mean
This work addresses the need for robust and private mean estimation in data analysis, particularly for applications requiring differential privacy, but it is incremental as it builds on existing subsample-and-aggregate methods.
The paper tackles the problem of designing a differentially private mean estimator for use in subsample-and-aggregate, introducing the private modified winsorized mean, which is shown to be minimax optimal for large classes of distributions and performs well empirically compared to other private estimators.
We develop a univariate, differentially private mean estimator, called the private modified winsorized mean, designed to be used as the aggregator in subsample-and-aggregate. We demonstrate, via real data analysis, that common differentially private multivariate mean estimators may not perform well as the aggregator, even in large datasets, motivating our developments.We show that the modified winsorized mean is minimax optimal for several, large classes of distributions, even under adversarial contamination. We also demonstrate that, empirically, the private modified winsorized mean performs well compared to other private mean estimates. We consider the modified winsorized mean as the aggregator in subsample-and-aggregate, deriving a finite sample deviations bound for a subsample-and-aggregate estimate generated with the new aggregator. This result yields two important insights: (i) the optimal choice of subsamples depends on the bias of the estimator computed on the subsamples, and (ii) the rate of convergence of the subsample-and-aggregate estimator depends on the robustness of the estimator computed on the subsamples.