Randomized Gradient-Free Distributed Online Optimization via a Dynamic Regret Analysis

This work considers 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 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.