We provide a general technique for making online learning algorithms differentially private, in both the full information and bandit settings. Our technique applies to algorithms that aim to minimize a \emph{convex} loss function which is a sum of smaller convex loss terms, one for each data point. We modify the popular \emph{mirror descent} approach, or rather a variant called \emph{follow the approximate leader}. The technique leads to the first nonprivate algorithms for private online learning in the bandit setting. In the full information setting, our algorithms improve over the regret bounds of previous work.

We give differentially private algorithms for a large class of online learning algorithms, in both the full information and bandit settings. Our algorithms aim to minimize a convex loss function which is a sum of smaller convex loss terms, one for each data point. To design our algorithms, we modify the popular mirror descent approach, or rather a variant called follow the approximate leader. The technique leads to the first nonprivate algorithms for private online learning in the bandit setting. In the full information setting, our algorithms improve over the regret bounds of previous work (due to Dwork, Naor, Pitassi and Rothblum (2010) and Jain, Kothari and Thakurta (2012)). In many cases, our algorithms (in both settings) match the dependence on the input length, T, of the optimal nonprivate regret bounds up to logarithmic factors in T. Our algorithms require logarithmic space and update time.

We consider two variants of private stochastic online learning. The first variant is differentially private stochastic bandits. Previously, Sajed and Sheffet (2019) devised the DP Successive Elimination (DP-SE) algorithm that achieves the optimal $ O \biggl(\sum\limits_{1\le j \le K: \Delta_j >0} \frac{ \log T}{ \Delta_j} + \frac{ K\log T}{\epsilon} \biggr)$ problem-dependent regret bound, where $K$ is the number of arms, $\Delta_j$ is the mean reward gap of arm $j$, $T$ is the time horizon, and $\epsilon$ is the required privacy parameter. However, like other elimination style algorithms, it is not an anytime algorithm. Until now, it was not known whether UCB-based algorithms could achieve this optimal regret bound. We present an anytime, UCB-based algorithm that achieves optimality. Our experiments show that the UCB-based algorithm is competitive with DP-SE. The second variant is the full information version of private stochastic online learning. Specifically, for the problem of decision-theoretic online learning with stochastic rewards, we present the first algorithm that achieves an $ O \left( \frac{ \log K}{ \Delta_{\min}} + \frac{\log(K) \min\{\log (\frac{1}{\Delta_{\min}}), \log(T)\}}{\epsilon} \right)$ regret bound, where $\Delta_{\min}$ is the minimum mean reward gap. In addition, we also show an $\Omega \left( \max\left\{ \frac{\log K}{\Delta_{\min}}, \frac{\log K}{\epsilon} \right\} \right)$ problem-dependent lower bound. The key idea behind our good theoretical guarantees in both settings is forgetfulness, i.e., decisions are made based on a certain amount of newly obtained observations instead of all the observations obtained from the very beginning.

We provide a general technique for making online learning algorithms differentially private, in both the full information and bandit settings. Our technique applies to algorithms that aim to minimize a \emph{convex} loss function which is a sum of smaller convex loss terms, one for each data point. We modify the popular \emph{mirror descent} approach, or rather a variant called \emph{follow the approximate leader}. The technique leads to the first nonprivate algorithms for private online learning in the bandit setting. In the full information setting, our algorithms improve over the regret bounds of previous work.

We give differentially private algorithms for a large class of online learning algorithms, inboth the full information and bandit settings. Our algorithms aim to minimize a convex loss function which is a sum of smaller convex loss terms, one for each data point. To design our algorithms, we modify the popular mirror descent approach, or rather a variant called follow the approximate leader. The technique leads to the first nonprivate algorithms for private online learning in the bandit setting. In the full information setting, our algorithms improve over the regret bounds of previous work (due to Dwork, Naor, Pitassi and Rothblum (2010) and Jain, Kothari and Thakurta (2012)). In many cases, our algorithms (in both settings) match the dependence on the input length, T, of the optimal nonprivate regret bounds up to logarithmic factors in T . Our algorithms require logarithmic space and update time.