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

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, $\mathtt{\text{ND-DFFP}}$ and $\mathtt{\text{NI-DFFP}}$, which integrate Nesterov-type acceleration with primal-dual techniques under two prominent settings: (i) $\mathit{\text{Lipschitz}}\mathit{\text{continuity}}\mathit{\text{of}}{G}_i\mathit{\text{and}}\mathit{\text{maximal}}\mathit{\text{monotonicity}}\mathit{\text{of}}{G}_i+T_i$; and (ii) $\mathit{\text{co-coercivity}}\mathit{\text{of}}{G}_i\mathit{\text{and}}\mathit{\text{maximal}}\mathit{\text{monotonicity}}\mathit{\text{of}}{T}_i$. While $\mathtt{\text{ND-DFFP}}$ utilizes a homogeneous network-dependent stepsize, $\mathtt{\text{NI-DFFP}}$ reformulates the problem into a three-operator inclusion to decouple the network topology, enabling heterogeneous network-independent stepsizes. Under appropriate assumptions, we establish an $\mathcal{O}(1\mathrm{/}k)$ convergence rate for the consensus error and an $\mathcal{O}(1\mathrm{/}k)$ 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.