  This work considered an online distributed optimization problem, with a group
of agents whose local objective functions vary with time. Moreover, the value
of the objective function is revealed to the corresponding agent after the
decision is executed per time-step. Thus, each agent can only update the
decision variable based on the revealed value and information collected from
the neighbors, without the knowledge on the explicit expression of the
objective function. To solve this problem, an online gradient-free distributed
projected gradient descent (DPGD) algorithm is presented, where each agent
locally approximates the gradient based on two point values. With some standard
assumptions on the communication graph and the objective functions, we provide
the bound for the dynamic regret as a function of the minimizer path length,
step-size and smoothing parameter. Under appropriate selections of the
step-size and smoothing parameter, we prove that the dynamic regret is
sublinear with respect to the time duration $T$ if the minimizer path length
also grows sublinearly. Finally, the effectiveness of the proposed algorithm is
illustrated through numerical simulations.
