Distributed Fast Fixed-Point Algorithms for Composite Monotone Inclusions over Networks

Abstract

This paper aims to develop new and efficient distributed algorithms for solving a class of monotone inclusions, $0\in {\sum }_{i=1}^n(G_{i}x+T_{i}x)$, over a connected network of $n$ agents, where the single-valued operator $G_i$ and the possibly multivalued operator $T_i$ remain private to agent $i$. Existing distributed algorithms for this problem class are primarily non-accelerated, and their exact convergence rates in the original primal space are largely unexplored. To bridge this gap, we propose two Decentralized Fast Fixed-Point-based algorithms, \texttt{ND-DFFP} and \texttt{NI-DFFP}, which integrate Nesterov-type acceleration with primal-dual techniques under two prominent settings: (i) \textit{Lipschitz continuity of $G_i$ and maximal monotonicity of $G_i+T_i$}; and (ii) \textit{co-coercivity of $G_i$ and maximal monotonicity of $T_i$}. While \texttt{ND-DFFP} utilizes a homogeneous network-dependent stepsize, \texttt{NI-DFFP} reformulates the problem into a three-operator inclusion to decouple the network topology, enabling heterogeneous network-independent stepsizes. Under appropriate assumptions, we establish an $\mathcal{O}(1/k)$ convergence rate for the consensus error and an $\mathcal{O}(1/k)$ rate for both the restricted gap function and the squared forward-backward splitting residual, with the latter two metrics evaluated at the network-average iterate or its projection onto the effective domain. Finally, numerical experiments on distributed bilinear matrix games and a virtual power plant problem demonstrate the competitive performance and computational efficiency of our methods over recent decentralized baselines in the literature.

71 pages, 6 tables, and 2 figures