  In this paper, we consider the problem of preserving privacy in the online
learning setting. We study the problem in the online convex programming (OCP)
framework---a popular online learning setting with several interesting
theoretical and practical implications---while using differential privacy as
the formal privacy measure. For this problem, we distill two critical
attributes that a private OCP algorithm should have in order to provide
reasonable privacy as well as utility guarantees: 1) linearly decreasing
sensitivity, i.e., as new data points arrive their effect on the learning model
decreases, 2) sub-linear regret bound---regret bound is a popular
goodness/utility measure of an online learning algorithm.
  Given an OCP algorithm that satisfies these two conditions, we provide a
general framework to convert the given algorithm into a privacy preserving OCP
algorithm with good (sub-linear) regret. We then illustrate our approach by
converting two popular online learning algorithms into their differentially
private variants while guaranteeing sub-linear regret ($O(\sqrt{T})$). Next, we
consider the special case of online linear regression problems, a practically
important class of online learning problems, for which we generalize an
approach by Dwork et al. to provide a differentially private algorithm with
just $O(\log^{1.5} T)$ regret. Finally, we show that our online learning
framework can be used to provide differentially private algorithms for offline
learning as well. For the offline learning problem, our approach obtains better
error bounds as well as can handle larger class of problems than the existing
state-of-the-art methods Chaudhuri et al.
