  Private regression has received attention from both database and security
communities. Recent work by Fredrikson et al. (USENIX Security 2014) analyzed
the functional mechanism (Zhang et al. VLDB 2012) for training linear
regression models over medical data. Unfortunately, they found that model
accuracy is already unacceptable with differential privacy when $\varepsilon =
5$. We address this issue, presenting an explicit connection between
differential privacy and stable learning theory through which a substantially
better privacy/utility tradeoff can be obtained. Perhaps more importantly, our
theory reveals that the most basic mechanism in differential privacy, output
perturbation, can be used to obtain a better tradeoff for all
convex-Lipschitz-bounded learning tasks. Since output perturbation is simple to
implement, it means that our approach is potentially widely applicable in
practice. We go on to apply it on the same medical data as used by Fredrikson
et al. Encouragingly, we achieve accurate models even for $\varepsilon = 0.1$.
In the last part of this paper, we study the impact of our improved
differentially private mechanisms on model inversion attacks, a privacy attack
introduced by Fredrikson et al. We observe that the improved tradeoff makes the
resulting differentially private model more susceptible to inversion attacks.
We analyze this phenomenon formally.
