Revisiting differentially private linear regression: optimal and adaptive prediction & estimation in unbounded domain

Linear regression is one of the oldest tools for data analysis (Galton, 1886) and it remains one of the most commonly-used as of today (Draper & Smith, 2014), especially in social sciences (Agresti & Finlay, 1997), econometics (Greene, 2003) and medical research (Armitage et al., 2008). Moreover, many nonlinear models are either intrinsically linear in certain function spaces, e.g., kernels methods, dynamical systems, or can be reduced to solving a sequence of linear regressions, e.g., iterative reweighted least square for generalized Linear models, gradient boosting for additive models and so on (see Friedman et al., 2001, for a detailed review). In order to apply linear regression to sensitive data such as those in social sciences and medical studies, it is often needed to do so such that the privacy of individuals in the data set is protected. Differential privacy (Dwork et al., 2006b) is a commonly-accepted criterion that provides provable protection against identification and is resilient to arbitrary auxiliary information that might be available to attackers. In this paper, we focus on linear regression with (ɛ, δ)-differentially privacy (Dwork et al., 2006a).