  This paper addresses safe distributed online optimization over an unknown set
of linear safety constraints. A network of agents aims at jointly minimizing a
global, time-varying function, which is only partially observable to each
individual agent. Therefore, agents must engage in local communications to
generate a safe sequence of actions competitive with the best minimizer
sequence in hindsight, and the gap between the two sequences is quantified via
dynamic regret. We propose distributed safe online gradient descent
(D-Safe-OGD) with an exploration phase, where all agents estimate the
constraint parameters collaboratively to build estimated feasible sets,
ensuring the action selection safety during the optimization phase. We prove
that for convex functions, D-Safe-OGD achieves a dynamic regret bound of
$O(T^{2/3} \sqrt{\log T} + T^{1/3}C_T^*)$, where $C_T^*$ denotes the
path-length of the best minimizer sequence. We further prove a dynamic regret
bound of $O(T^{2/3} \sqrt{\log T} + T^{2/3}C_T^*)$ for certain non-convex
problems, which establishes the first dynamic regret bound for a safe
distributed algorithm in the non-convex setting.
