  We study the online saddle point problem, an online learning problem where at
each iteration a pair of actions need to be chosen without knowledge of the
current and future (convex-concave) payoff functions. The objective is to
minimize the gap between the cumulative payoffs and the saddle point value of
the aggregate payoff function, which we measure using a metric called
"SP-Regret". The problem generalizes the online convex optimization framework
but here we must ensure both players incur cumulative payoffs close to that of
the Nash equilibrium of the sum of the games. We propose an algorithm that
achieves SP-Regret proportional to $\sqrt{\ln(T)T}$ in the general case, and
$\log(T)$ SP-Regret for the strongly convex-concave case. We also consider the
special case where the payoff functions are bilinear and the decision sets are
the probability simplex. In this setting we are able to design algorithms that
reduce the bounds on SP-Regret from a linear dependence in the dimension of the
problem to a \textit{logarithmic} one. We also study the problem under bandit
feedback and provide an algorithm that achieves sublinear SP-Regret. We then
consider an online convex optimization with knapsacks problem motivated by a
wide variety of applications such as: dynamic pricing, auctions, and
crowdsourcing. We relate this problem to the online saddle point problem and
establish $O(\sqrt{T})$ regret using a primal-dual algorithm.
