  We study the problem of safe online convex optimization, where the action at
each time step must satisfy a set of linear safety constraints. The goal is to
select a sequence of actions to minimize the regret without violating the
safety constraints at any time step (with high probability). The parameters
that specify the linear safety constraints are unknown to the algorithm. The
algorithm has access to only the noisy observations of constraints for the
chosen actions. We propose an algorithm, called the {Safe Online Projected
Gradient Descent} (SO-PGD) algorithm, to address this problem. We show that,
under the assumption of the availability of a safe baseline action, the SO-PGD
algorithm achieves a regret $O(T^{2/3})$. While there are many algorithms for
online convex optimization (OCO) problems with safety constraints available in
the literature, they allow constraint violations during learning/optimization,
and the focus has been on characterizing the cumulative constraint violations.
To the best of our knowledge, ours is the first work that provides an algorithm
with provable guarantees on the regret, without violating the linear safety
constraints (with high probability) at any time step.
