  This paper studies a distributed online constrained optimization problem over
time-varying unbalanced digraphs without explicit subgradients. In sharp
contrast to the existing algorithms, we design a novel consensus-based
distributed online algorithm with a local randomized zeroth-order oracle and
then rescale the oracle by constructing row-stochastic matrices, which aims to
address the unbalancedness of time-varying digraphs. Under mild conditions, the
average dynamic regret over a time horizon is shown to asymptotically converge
at a sublinear rate provided that the accumulated variation grows sublinearly
with a specific order. Moreover, the counterpart of the proposed algorithm when
subgradients are available is also provided, along with its dynamic regret
bound, which reflects that the convergence of our algorithm is essentially not
affected by the zeroth-order oracle. Simulations on distributed targets
tracking problem and dynamic sparse signal recovery problem in sensor networks
are employed to demonstrate the effectiveness of the proposed algorithm.
