  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 problems
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}{\epsilon} \right)$ regret bound, where $\Delta_{\min}$ is the
minimum mean reward gap. The key idea behind our good theoretical guarantees in
both settings is the 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.
