Distributed Learning for Online Stochastic Mixed Equilibrium Problems With Heavy-Tailed Noises

In this article, mixed equilibrium problems (MEPs) in uncertain and dynamic environments are investigated by employing a multiagent system (MAS), where each agent only has access to its own bifunction, and can communicate with its immediate neighbors via a time-varying digraph. At each time, the goal of agents is to cooperatively find a point in the constraint set such that the sum of local bifunctions with a free variable is nonnegative. Different from existing works on MEPs, here the bifunctions are stochastic and time-varying, and only available to agents after decisions are made. To tackle this problem, an online distributed learning algorithm based on the mirror descent algorithm and the clipping strategy is proposed. The dynamic regret, whose offline benchmark is to find the solution at each time, is employed to measure the performance of the algorithm. Of particular interest is that the high probability bound of the regret is established under the heavy-tailed noise condition. The result shows that if the variation in the solution sequence is within a certain range, then the dynamic regret grows sublinearly with high probability. Finally, a simulation example is provided to corroborate the validity of the theoretical results.