  We consider the problem of online classification under a privacy constraint.
In this setting a learner observes sequentially a stream of labelled examples
$(x_t, y_t)$, for $1 \leq t \leq T$, and returns at each iteration $t$ a
hypothesis $h_t$ which is used to predict the label of each new example $x_t$.
The learner's performance is measured by her regret against a known hypothesis
class $\mathcal{H}$. We require that the algorithm satisfies the following
privacy constraint: the sequence $h_1, \ldots, h_T$ of hypotheses output by the
algorithm needs to be an $(\epsilon, \delta)$-differentially private function
of the whole input sequence $(x_1, y_1), \ldots, (x_T, y_T)$. We provide the
first non-trivial regret bound for the realizable setting. Specifically, we
show that if the class $\mathcal{H}$ has constant Littlestone dimension then,
given an oblivious sequence of labelled examples, there is a private learner
that makes in expectation at most $O(\log T)$ mistakes -- comparable to the
optimal mistake bound in the non-private case, up to a logarithmic factor.
Moreover, for general values of the Littlestone dimension $d$, the same mistake
bound holds but with a doubly-exponential in $d$ factor. A recent line of work
has demonstrated a strong connection between classes that are online learnable
and those that are differentially-private learnable. Our results strengthen
this connection and show that an online learning algorithm can in fact be
directly privatized (in the realizable setting). We also discuss an adaptive
setting and provide a sublinear regret bound of $O(\sqrt{T})$.
