  In this paper, we consider two paradigms that are developed to account for
uncertainty in optimization models: robust optimization (RO) and joint
estimation-optimization (JEO). We examine recent developments on efficient and
scalable iterative first-order methods for these problems, and show that these
iterative methods can be viewed through the lens of online convex optimization
(OCO). The standard OCO framework has seen much success for its ability to
handle decision-making in dynamic, uncertain, and even adversarial
environments. Nevertheless, our applications of interest present further
flexibility in OCO via three simple modifications to standard OCO assumptions:
we introduce two new concepts of weighted regret and online saddle point
problems and study the possibility of making lookahead (anticipatory)
decisions. Our analyses demonstrate that these flexibilities introduced into
the OCO framework have significant consequences whenever they are applicable.
For example, in the strongly convex case, minimizing unweighted regret has a
proven optimal bound of $O(\log(T)/T)$, whereas we show that a bound of
$O(1/T)$ is possible when we consider weighted regret. Similarly, for the
smooth case, considering $1$-lookahead decisions results in a $O(1/T)$ bound,
compared to $O(1/\sqrt{T})$ in the standard OCO setting. Consequently, these
OCO tools are instrumental in exploiting structural properties of functions and
resulting in improved convergence rates for RO and JEO. In certain cases, our
results for RO and JEO match the best known or optimal rates in the
corresponding problem classes without data uncertainty.
