Regret bounds of a distributed saddle point algorithm

An algorithm to learn optimal actions in distributed convex repeated games is developed. Learning is repeated because cost functions are revealed sequentially and distributed because they are revealed to agents of a network that can exchange information with neighboring nodes only. Learning is measured in terms of the global networked regret, which is the accumulated loss of causal prediction with respect to a centralized clairvoyant agent to which the information of all times and agents is revealed at the initial time. We use a variant of the Arrow-Hurwicz saddle point algorithm which penalizes local agent disagreement via Lagrange multipliers and leads to a distributed online algorithm. We show that decisions made with this saddle point algorithm lead to regret whose order is not larger than O(√T), where T is the total number of rounds of the game. Numerical behavior is illustrated for the particular case of dynamic sensor network estimation across different network sizes, connectivities, and topologies.