Distributed Forgetting-factor Regret-based Online Optimization over Undirected Connected Networks

The evaluation of final-iteration tracking performance is a formidable obstacle in distributed online optimization algorithms. To address this issue, this paper proposes a novel evaluation metric named distributed forgetting-factor regret (DFFR). It incorporates a weight into the loss function at each iteration, which progressively reduces the weights of historical loss functions while enabling dynamic weights allocation across optimization horizon. Furthermore, we develop two distributed online optimization algorithms based on DFFR over undirected connected networks: the Distributed Online Gradient-free Algorithm for bandit-feedback problems and the Distributed Online Projection-free Algorithm for high-dimensional problems. Through theoretical analysis, we derive the upper bounds of DFFR for both algorithms and further prove that under mild conditions, DFFR either converges to zero or maintains a tight upper bound as iterations approach infinity. Experimental simulation demonstrates the effectiveness of the algorithms and the superior performance of DFFR.