arXiv subjects

Mauro Franceschelli

Publications and source records attributed to Mauro Franceschelli.

16 recordsLinked to original sources

arXiv2025

Non-Euclidean Enriched Contraction Theory for Monotone Operators and Monotone Dynamical Systems

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.

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arXiv2025

Optimization and Learning in Open Multi-Agent Systems

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 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 to 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, allowing to characterize their performance in terms of the punctual distance from 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 or tracking problems on different metrics of interest, such as average, median, and min/max value, as well as classification problems with logistic loss functions.

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arXiv2024

Algebraic Connectivity Control and Maintenance in Multi-Agent Networks under Attack

This paper studies the problem of increasing the connectivity of an ad-hoc peer-to-peer network subject to cyber-attacks targeting the agents in the network. The adopted strategy involves the design of local interaction rules for the agents to locally modify the graph topology by adding and removing links with neighbors. Two distributed protocols are presented to boost the algebraic connectivity of the network graph beyond $k-2\sqrt{k-1}$ where $k\in \mathbb{N}$ is a free design parameter; these two protocols are achieved through the distributed construction of random (approximate) regular graphs. One protocol leverages coordinated actions between pairs of neighboring agents and is mathematically proven to converge to the desired graph topology. The other protocol relies solely on the uncoordinated actions of individual agents and it is validated by a spectral analysis through Monte-Carlo simulations. Numerical simulations offer a comparative analysis with other state-of-the-art algorithms, showing the ability of both proposed protocols to maintain high levels of connectivity despite attacks carried out with full knowledge of the network structure, and highlighting their superior performance.

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arXiv2024

Accelerated Alternating Direction Method of Multipliers Gradient Tracking for Distributed Optimization

This paper presents a novel accelerated distributed algorithm for unconstrained consensus optimization over static undirected networks. The proposed algorithm combines the benefits of acceleration from momentum, the robustness of the alternating direction method of multipliers, and the computational efficiency of gradient tracking to surpass existing state-of-the-art methods in convergence speed, while preserving their computational and communication cost. First, we prove that, by applying momentum on the average dynamic consensus protocol over the estimates and gradient, we can study the algorithm as an interconnection of two singularly perturbed systems: the outer system connects the consensus variables and the optimization variables, and the inner system connects the estimates of the optimum and the auxiliary optimization variables. Next, we prove that, by adding momentum to the auxiliary dynamics, our algorithm always achieves faster convergence than the achievable linear convergence rate for the non-accelerated alternating direction method of multipliers gradient tracking algorithm case. Through simulations, we numerically show that our accelerated algorithm surpasses the existing accelerated and non-accelerated distributed consensus first-order optimization protocols in convergence speed.

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arXiv2023

Robust Online Learning over Networks

The recent deployment of multi-agent 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: (i) online training, i.e., the local data change over time; (ii) asynchronous agent computations; (iii) unreliable and limited communications; and (iv) 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 (i)-(iv). 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 (i)-(iv).

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arXiv2023

Novel Stability Conditions for Nonlinear Monotone Systems and Consensus in Multi-Agent Networks

In this work, we characterize a class of nonlinear monotone dynamical systems that have a certain translation invariance property which goes by the name of plus-homogeneity; usually called "topical" systems. Such systems need not be asymptotically stable, since they are merely nonexpansive but not contractive. Thus, we introduce a stricter version of monotonicity, termed "type-K" in honor of Kamke, and we prove the asymptotic stability of the equilibrium points, as well as the convergence of all trajectories to such equilibria for type-K monotone and plus-homogeneous systems: we call them "K-topical". Since topical maps are the natural nonlinear counterpart of linear maps defined by row-stochastic matrices, which are a cornerstone in the convergence analysis of linear multi-agent systems (MASs), we exploit our results for solving the consensus problem over nonlinear K-topical MASs. We first provide necessary and sufficient conditions on the local interaction rules of the agents ensuring the K-topicality of a MAS. Then, we prove that the agents achieve consensus asymptotically if the graph describing their interactions contains a globally reachable node. Finally, several examples for continuous-time and discrete-time systems are discussed to corroborate the enforceability of our results in different applications.

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arXiv2020

Dynamic Max-Consensus and Size Estimation of Anonymous Multi-Agent Networks

In this paper we propose a novel consensus protocol for discrete-time multi-agent systems (MAS), which solves the dynamic consensus problem on the max value, i.e., the dynamic max-consensus problem. In the dynamic max-consensus problem to each agent is fed a an exogenous reference signal, the objective of each agent is to estimate the instantaneous and time-varying value of the maximum among the signals fed to the network, by exploiting only local and anonymous interactions among the agents. The absolute and relative tracking error of the proposed distributed control protocol is theoretically characterized and is shown to be bounded and by tuning its parameters it is possible to trade-off convergence time for steady-state error. The dynamic Max-consensus algorithm is then applied to solve the distributed size estimation problem in a dynamic setting where the size of the network is time-varying during the execution of the estimation algorithm. Numerical simulations are provided to corroborate the theoretical analysis.

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arXiv2019

Multi-Agent Coordination of Thermostatically Controlled Loads by Smart Power Sockets for Electric Demand Side Management

This paper presents a multi-agent control architecture and an online optimization method based on dynamic average consensus to coordinate the power consumption of a large population of Thermostatically Controlled Loads (TCLs). Our objective is to penalize peaks of power demand, smooth the load profile and enable Demand Side Management (DSM). The proposed architecture and methods exploit only local measurements of power consumption via Smart Power Sockets (SPSs) with no access to their internal temperature. No centralized aggregator of information is exploited and agents preserve their privacy by cooperating anonymously only through consensus-based distributed estimation, robust to node/link failure. The interactions among devices are designed to occur through an unstructured peer-to-peer (P2P) network over the internet. The architecture includes novel methods for parameter identification, state estimation and mixed logical modelling of TCLs and SPSs. It is designed from a multi-agent and plug-and-play perspective in which existing household appliances can interact with each other in an urban environment. Finally, a novel low cost testbed is proposed along with numerical tests and an experimental validation.

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arXiv2019

A Nonlinear Perron-Frobenius Approach for Stability and Consensus of Discrete-Time Multi-Agent Systems

In this paper we propose a novel method to establish stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) evolving according to nonlinear heterogeneous local interaction rules which is not based on Lyapunov function arguments. In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by sub-homogeneous and order-preserving nonlinear maps. This paper directly generalizes results for sub-homogeneous and order-preserving linear maps which are shown to be the counterpart to stochastic matrices thanks to nonlinear Perron-Frobenius theory. We provide sufficient conditions on the structure of local interaction rules among agents to establish convergence to a fixed point and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the method are provided to corroborate the theoretical analysis.

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arXiv2019

Stability of Open Multi-Agent Systems and Applications to Dynamic Consensus

In this technical note we consider a class of multi-agent network systems that we refer to as Open Multi-Agent Systems (OMAS): in these multi-agent systems, an indefinite number of agents may join or leave the network at any time. Focusing on discrete-time evolutions of scalar agents, we provide a novel theoretical framework to study the dynamical properties of OMAS: specifically, we propose a suitable notion of stability and derive sufficient conditions to ensure stability in this sense. These sufficient conditions regard the arrival/departure of an agent as a disturbance: consistently, they require the effect of arrivals/departures to be bounded (in a precise sense) and the OMAS to be contractive in the absence of arrivals/departures. In order to provide an example of application for this theory, we re-formulate the well-known Proportional Dynamic Consensus for Open Multi-Agent Systems and we characterize the stability properties of the resulting Open Proportional Dynamic Consensus algorithm.

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arXiv2017

A duality-based approach for distributed min-max optimization with application to demand side management

In this paper we consider a distributed optimization scenario in which a set of processors aims at minimizing the maximum of a collection of "separable convex functions" subject to local constraints. This set-up is motivated by peak-demand minimization problems in smart grids. Here, the goal is to minimize the peak value over a finite horizon with: (i) the demand at each time instant being the sum of contributions from different devices, and (ii) the local states at different time instants being coupled through local dynamics. The min-max structure and the double coupling (through the devices and over the time horizon) makes this problem challenging in a distributed set-up (e.g., well-known distributed dual decomposition approaches cannot be applied). We propose a distributed algorithm based on the combination of duality methods and properties from min-max optimization. Specifically, we derive a series of equivalent problems by introducing ad-hoc slack variables and by going back and forth from primal and dual formulations. On the resulting problem we apply a dual subgradient method, which turns out to be a distributed algorithm. We prove the correctness of the proposed algorithm and show its effectiveness via numerical computations.

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arXiv2016

A duality-based approach for distributed min-max optimization

In this paper we consider a distributed optimization scenario in which a set of processors aims at cooperatively solving a class of min-max optimization problems. This set-up is motivated by peak-demand minimization problems in smart grids. Here, the goal is to minimize the peak value over a finite horizon with: (i) the demand at each time instant being the sum of contributions from different devices, and (ii) the device states at different time instants being coupled through local constraints (e.g., the dynamics). The min-max structure and the double coupling (through the devices and over the time horizon) makes this problem challenging in a distributed set-up (e.g., existing distributed dual decomposition approaches cannot be applied). We propose a distributed algorithm based on the combination of duality methods and properties from min-max optimization. Specifically, we repeatedly apply duality theory and properly introduce ad-hoc slack variables in order to derive a series of equivalent problems. On the resulting problem we apply a dual subgradient method, which turns out to be a distributed algorithm consisting of a minimization on the original primal variables and a suitable dual update. We prove the convergence of the proposed algorithm in objective value. Moreover, we show that every limit point of the primal sequence is an optimal (feasible) solution. Finally, we provide numerical computations for a peak-demand optimization problem in a network of thermostatically controlled loads.

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arXiv2015

Consensus on the average in arbitrary directed network topologies with time-delays

In this preliminary paper we study the stability property of a consensus on the average algorithm in arbitrary directed graphs with respect to communication/sensing time-delays. The proposed algorithm adds a storage variable to the agents' states so that the information about the average of the states is preserved despite the algorithm iterations are performed in an arbitrary strongly connected directed graph. We prove that for any network topology and choice of design parameters the consensus on the average algorithm is stable for sufficiently small delays. We provide simulations and numerical results to estimate the maximum delay allowed by an arbitrary unbalanced directed network topology.

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arXiv2014

Consensus in multi-agent systems with non-periodic sampled-data exchange and uncertain network topology

In this paper consensus in second-order multi-agent systems with a non-periodic sampled-data exchange among agents is investigated. The sampling is random with bounded inter-sampling intervals. It is assumed that each agent has exact knowledge of its own state at any time instant. The considered local interaction rule is PD-type. Sufficient conditions for stability of the consensus protocol to a time-invariant value are derived based on LMIs. Such conditions only require the knowledge of the connectivity of the graph modeling the network topology. Numerical simulations are presented to corroborate the theoretical results.

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arXiv2014

Consensus in multi-agent systems with second-order dynamics and non-periodic sampled-data exchange

In this paper consensus in second-order multi-agent systems with a non-periodic sampled-data exchange among agents is investigated. The sampling is random with bounded inter-sampling intervals. It is assumed that each agent has exact knowledge of its own state at all times. The considered local interaction rule is PD-type. The characterization of the convergence properties exploits a Lyapunov-Krasovskii functional method, sufficient conditions for stability of the consensus protocol to a time-invariant value are derived. Numerical simulations are presented to corroborate the theoretical results.

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arXiv2012

Decentralized Estimation of Laplacian Eigenvalues in Multi-Agent Systems

In this paper we present a decentralized algorithm to estimate the eigenvalues of the Laplacian matrix that encodes the network topology of a multi-agent system. We consider network topologies modeled by undirected graphs. The basic idea is to provide a local interaction rule among agents so that their state trajectory is a linear combination of sinusoids oscillating only at frequencies function of the eigenvalues of the Laplacian matrix. In this way, the problem of decentralized estimation of the eigenvalues is mapped into a standard signal processing problem in which the unknowns are the finite number of frequencies at which the signal oscillates.

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