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

This note 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 first derive an upper bound on the dynamic regret due to each agent, as a product of a sublinear function of the time duration $T$ and the deviation of the minimizer sequence. Then, with an appropriate selection of the step-size sequence, we are able to establish a regret bound of $\mathcal{O}(1/\sqrt{T})$ when the variation of the minimizer sequence is sublinear. The effectiveness of the proposed algorithm is illustrated through numerical simulations.