Distributed Online Stochastic Convex-Concave Optimization: Dynamic Regret Analyses under Single and Multiple Consensus Steps

TL;DR

This paper introduces a distributed online stochastic mirror descent algorithm for convex-concave optimization over multiagent networks, achieving sublinear dynamic regret bounds and improved convergence with multiple consensus steps, validated through simulations.

Contribution

It proposes a novel distributed stochastic mirror descent algorithm with time-varying predictive mappings and analyzes its dynamic regret bounds under various consensus strategies.

Findings

Achieves dynamic regret upper-bound of O(max{T^θ1, T^θ2}(1+V_T))

Guarantees sublinear convergence when path-variation V_T is sublinear

Enhanced regret bounds using multiple consensus technique

Abstract

This paper considers the distributed online convex-concave optimization with constraint sets over a multiagent network, in which each agent autonomously generates a series of decision pairs through a designable mechanism to cooperatively minimize the global loss function. To this end, under no-Euclidean distance metrics, we propose a distributed online stochastic mirror descent convex-concave optimization algorithm with time-varying predictive mappings. Taking dynamic saddle point regret as a performance metric, it is proved that the proposed algorithm achieves the regret upper-bound in $\mathcal{O}(\max \{T^{\theta_1}, T^{\theta_2} (1+V_T ) \})$ for the general convex-concave loss function, where $\theta_1, \theta_2 \in(0,1)$ are the tuning parameters, $T$ is the total iteration time, and $V_T$ is the path-variation. Surely, this algorithm guarantees the sublinear convergence, provided that $V_T$ is sublinear. Moreover, aiming to achieve better convergence, we further investigate a variant of this algorithm by employing the multiple consensus technique. The obtained results show that the appropriate setting can effectively tighten the regret bound to a certain extent. Finally, the efficacy of the proposed algorithms is validated and compared through the simulation example of a target tracking problem.