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

Dynamic Regret Analysis of Safe Distributed Online Optimization for Convex and Non-convex Problems