Near-Optimal Algorithms for Private Online Learning in a Stochastic Environment

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.