Time-varying multi-objective optimisation over switching graphs via fixed-time consensus algorithms

Abstract

This paper considers distributed multi-objective optimisation problems with time-varying cost functions for network connected multi-agent systems over switching graphs. The scalarisation approach is used to convert the problem into a weighted-sum objective. Fixed-time consensus algorithms are developed for each agent to estimate the global variables, and drive all local copies of the decision vector to a consensus. The algorithm with fixed gains is first proposed, where some global information is required to choose the gains. Then, an adaptive algorithm is presented to eliminate the use of global information. The convergence of those algorithms to the Pareto solutions is established via Lyapunov theory for connected graphs. In case of disconnected graphs, the convergence to the subsets of the Pareto fronts is studied. Simulation results are provided to demonstrate the effectiveness of the proposed algorithms.

Distributed Support Vector Machines Over Dynamic Balanced Directed Networks

“…where the last inequality follows the strict convexity of the loss function (see Lemma 3). Recall that from Lemma 5 the derivatives dλ1,j dα | α=0 and dλ2,j dα | α=0 depend on the eigenvalues of (15), which clearly form a lower triangular matrix with m zero eigenvalues and m negative eigenvalues (following ( 16)). Therefore, dλ1,j dα | α=0 = 0 and dλ2,j dα | α=0 < 0, which implies that considering αM 1 as a perturbation, the m zero eigenvalues λ 2,j (α) of M move toward the LHP while λ 1,j (α)'s remain zero.…”

“…which completes the proof. Theorem 1, similar to [10]- [15], only requires strict convexity of the loss function, as compared to strong convexity in [20]- [26]. Moreover, the matrix perturbation method allows eigen-spectrum analysis of the time-varying matrix M , including possible discrete jumps in the hybrid mode.…”

“…To analyze the proposed hybrid model, we use matrix perturbation theory [31] to char-acterize the eigen-spectrum of the proposed dynamics, which enables convergence analysis in the hybrid DT-CT setup. Due to Lipschitz-continuity of the proposed CT approach, it's DT approximation is free of the aforementioned chattering inherent to the non-Lipschitz dynamics [11]- [15]. Further, the proposed solution is free of penalty-based approximation inaccuracies in [17], [18], [27].…”

“…A double time-scale algorithm is proposed by [20] with finite number of communications per gradient-update iteration. Other methods include finite/fixed-time algorithms [11]- [15] that are prone to steady-state chattering due to non-Lipschitz dynamics.…”

On the game‐theoretic analysis of distributed generative adversarial networks

“…Distributed learning over network‐connected multiagent systems has attracted significant research interests, due to its wide applications in the fields of social science, economy and engineering 6–10 . Recently, there have been a number of pioneering works dedicated to constructing distributed learning frameworks using consensus‐based approaches, for example, References [11–14].…”