Diego Deplano

This paper studies distributed composite optimization in open multi-agent systems, where agents may join and leave the network over time. We propose a variant of Open ADMM in which the composite proximal step is approximated through a finite number of proximal-gradient iterations, enabling agents to handle composite objectives even when the exact proximal operator is not available in closed form. Since the inexact computation of the proximal step induces additive errors, we establish a convergence result for paracontractive open multi-agent systems subject to bounded additive perturbations that capture both inexact local updates and noisy communications. Under mild assumptions on the network topology, the proposed algorithm converges linearly to a neighborhood of the optimal consensus set, with an explicit bound on the steady-state tracking error. The theoretical guarantees are corroborated by numerical experiments on distributed learning applications, namely linear regression (supervised learning) and principal component analysis (unsupervised learning).

Modern artificial intelligence relies on networks of agents that collect data, process information, and exchange it with neighbors to collaboratively solve optimization and learning problems. This article introduces a novel distributed algorithm to address a broad class of these problems in open networks, where the number of participating agents may vary due to several factors, such as autonomous decisions, heterogeneous resource availability, or denial of service (DoS) attacks. Extending the current literature, the convergence analysis of the proposed algorithm is based on the newly developed Theory of Open Operators, which characterizes an operator as open when the set of components to be updated changes over time, yielding time-varying operators acting on sequences of points of different dimensions and compositions. The mathematical tools and convergence results developed here provide a general framework for evaluating distributed algorithms in open networks, enabling characterization of their performance in terms of the punctual distance to the optimal solution, in contrast with regret-based metrics that assess cumulative performance over a finite-time horizon. As illustrative examples, the proposed algorithm is used to solve dynamic consensus and tracking problems on different metrics, such as average, median, and min/max, as well as classification problems with logistic loss functions.

We present a distributed algorithm enabling the dynamic tracking of diameter and radius (DR) of time-varying open networks (ON) together with nodes' eccentricities and network size (ES), called $\rm {DR-ON-ES}$, which does not require the disclosure of nodes' identity nor any a priori information on the network. The convergence properties of $\rm {DR-ON-ES}$ are discussed within the framework of open multi-agent systems (OMASs): sufficient conditions are provided to ensure that the TPI of an OMAS executing $\rm {DR-ON-ES}$ is globally asymptotically open stable, meaning that all trajectories converge to within a neighborhood of the sought parameters. The parameters' estimations at equilibrium points are characterized in terms of mean expected value and mean squared error. As an ancillary result, we remark that $\rm {DR-ON-ES}$ exploits the OSTDMC Protocol – a novel distributed protocol formalized and analyzed in this manuscript – to achieve consensus on the time-varying maximum value of a set of signals fed locally to the agents of the network.

We adopt an operator-theoretic perspective to analyze a class of nonlinear fixed-point iterations and discrete-time dynamical systems. Specifically, we study the Krasnoselskij iteration - at the heart of countless algorithmic schemes and underpinning the stability analysis of numerous dynamical models - by focusing on a non-Euclidean vector space equipped with the diagonally weighted supremum norm. By extending the state of the art, we introduce the notion of enriched weak contractivity, which (i) is characterized by a simple, verifiable condition for Lipschitz operators, and (ii) yields explicit bounds on the admissible step size for the Krasnoselskij iteration. Our results relate the notion of weak contractivity with that of monotonicity of operators and dynamical systems and show its generality to design larger step sizes and improved convergence speed for broader classes of dynamical systems. The newly developed theory is illustrated on two applications: the design of zero-finding algorithms for monotone operators and the design of nonlinear consensus dynamics in monotone multi-agent dynamical systems.

The recent deployment of multiagent networks has enabled the distributed solution of learning problems, where agents cooperate to train a global model without sharing their local, private data. This work specifically targets some prevalent challenges inherent to distributed learning: 1) online training, i.e., the local data change over time; 2) asynchronous agent computations; 3) unreliable and limited communications; and 4) inexact local computations. To tackle these challenges, we apply the distributed operator theoretical (DOT) version of the alternating direction method of multipliers (ADMM), which we call “DOT-ADMM.” We prove that if the DOT-ADMM operator is metric subregular, then it converges with a linear rate for a large class of (not necessarily strongly) convex learning problems toward a bounded neighborhood of the optimal time-varying solution, and characterize how such neighborhood depends on 1)–4). We first derive an easy-to-verify condition for ensuring the metric subregularity of an operator, followed by tutorial examples on linear and logistic regression problems. We corroborate the theoretical analysis with numerical simulations comparing DOT-ADMM with other state-of-the-art algorithms, showing that only the proposed algorithm exhibits robustness to 1)–4).

This letter shows that trajectories of continuous-time monotone systems (in the sense of Kamke-Muller) converge to equilibrium points if their vector field is continuously differentiable and if they are nonexpansive w.r.t. a diagonally weighted infinity norm. Differently from the current literature trend, the system is not required to be contractive but merely nonexpansive, thus allowing for multiple equilibrium points. Easy-to-check conditions on the vector field to verify that the system is both monotone and nonexpansive are provided. This is done by showing that nonexpansiveness is implied by subhomogeneity of the system, a generalization of the translation invariance property. We apply the results in the context of RNNs, thus providing sufficient conditions for convergence of the state trajectories of nonexpansive monotone neural networks that are not contractive.