Median regression with differential privacy

Personal privacy information may be exposed with the unprecedented availability of datasets, so there is increasing requirement that statistical analysis of such datasets should protect individual privacy. As [6] describes, differential privacy addresses the paradox of learning nothing about an individual while learning useful information about a population. Over the past few years, differential privacy has been investigated in machine learning [1] and has been applied in the real world, see for example [8]. Recently, [3] formulates a general lower bound argument for minimax risks with differential privacy constraints, and applies this argument to high-dimensional mean estimation and linear regression problems. In this paper, three privacy preserving methods are proposed for median regression, which is a special case of quantile regression. Quantile regression was first introduced in [12], which aims to estimate and conduct inference about conditional quantile functions. In recent years, quantile regression has become a comprehensive method for statistical analysis of response models and it has been widely used in reality, such as survival analysis and economics, see for example, [14], [20] and [15]. The fact that the median regression takes least absolute deviation as its objective function to estimate parameters has been known among statisticians [12].