  We propose a new framework to solve online optimization and learning problems
in unknown and uncertain dynamical environments. This framework enables us to
simultaneously learn the uncertain dynamical environment while making online
decisions in a quantifiably robust manner. The main technical approach relies
on the theory of distributional robust optimization that leverages adaptive
probabilistic ambiguity sets. However, as defined, the ambiguity set usually
leads to online intractable problems, and the first part of our work is
directed to find reformulations in the form of online convex problems for two
sub-classes of objective functions. To solve the resulting problems in the
proposed framework, we further introduce an online version of the Nesterov
accelerated-gradient algorithm. We determine how the proposed solution system
achieves a probabilistic regret bound under certain conditions. Two
applications illustrate the applicability of the proposed framework.
