A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

Bin, Michelangelo; Notarnicola, Ivano; Marconi, Lorenzo; Notarstefano, Giuseppe

doi:10.1109/cdc40024.2019.9029824

Cited by 22 publications

(16 citation statements)

References 18 publications

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“…We start by first introducing an alternative, causal formulation of the (discrete) gradient tracking that has been proposed in [39]. At each iteration k ∈ N, each agent i maintains a local estimate x i,k ∈ R d of the optimal solution of problem (1) and an auxiliary state z i,k ∈ R d that are updated according to…”

Section: A From Discrete To Continuousmentioning

confidence: 99%

“…There exist several variants of the gradient tracking algorithm, see [29]- [38]. A control-based analysis of this algorithm has been proposed in [39], in which a suitable set of coordinates is considered which turns out to be fundamental for the analysis performed in this paper. Finally, the approach in [39] has been exploited to design gradient tracking algorithms with sparse (non-necessarily diagonal) gains in [40].…”

Section: Introductionmentioning

confidence: 99%

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Triggered Gradient Tracking for Asynchronous Distributed Optimization

Carnevale¹,

Notarnicola²,

Marconi³

et al. 2022

Preprint

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This paper proposes ASYNCHRONOUS TRIGGERED GRADIENT TRACKING, i.e., a distributed optimization algorithm to solve consensus optimization over networks with asynchronous communication. As a building block, we start by devising the continuous-time counterpart of the recently proposed (discretetime) distributed gradient tracking called CONTINUOUS GRA-DIENT TRACKING. By using a Lyapunov approach, we prove exponential stability of the equilibrium corresponding to agents' estimates being consensual to the optimal solution of the problem, with arbitrary initialization of the local estimates. Then, we propose two triggered versions of the algorithm. In the first one, the agents continuously integrate their local dynamics and exchange with neighbors current local variables according to a synchronous communication protocol. In ASYNCHRONOUS TRIGGERED GRADIENT TRACKING, we propose a totally asynchronous scheme in which each agent sends to neighbors its current local variables based on a triggering condition that depends on a locally verifiable condition. The triggering protocol preserves the linear convergence of the algorithm and excludes the Zeno behavior. By using the stability analysis of CONTINUOUS GRADIENT TRACKING as a preparatory result, we show exponential stability of the equilibrium point holds for both triggered algorithms and any estimates' initialization. Finally, numerical simulations validate the effectiveness of the proposed methods on a data analytics problem, showing also the improved performance in terms of inter-agent communication.

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Section: A From Discrete To Continuousmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Triggered Gradient Tracking for Asynchronous Distributed Optimization

Carnevale¹,

Notarnicola²,

Marconi³

et al. 2022

Preprint

Self Cite

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“…Then, to have system (5) in a strictly causal form, we use a change of coordinates similar to the one proposed in [38], i.e.,…”

Section: Extremum Seeking Tracking: Stability Analysismentioning

confidence: 99%

Extremum Seeking Tracking for Derivative-free Distributed Optimization

Mimmo,

Carnevale,

Testa

et al. 2021

Preprint

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In this paper, we deal with a network of agents that want to cooperatively minimize the sum of local cost functions depending on a common decision variable. We consider the challenging scenario in which objective functions are unknown and agents have only access to local measurements of their local function. We propose a novel distributed algorithm which combines a gradient tracking policy with an extremum seeking technique to estimate the global descent direction. The joint use of these two techniques results in a distributed optimization scheme which is proven to provide arbitrarily accurate solution estimate through the combination of Lyapunov and averaging analysis approaches with consensus theory. To corroborate the theoretical results, we perform numerical simulations in a personalized optimization framework and a cooperative robotics scenario.

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“…If the minimum sharing problem is cast in terms of the optimization problem (2), then one can rely on a welldeveloped literature on discrete-time distributed optimization (see [1] for a recent overview). If the functions ψ i in (2) are convex, indeed, different approaches can be used, such as consensus-based (sub)gradient methods [11][12][13][14][15][16], second-order methods [17,18], projected [19] and primaldual [20,21] methods with inequality constraints, methods based on the distributed Alternate Direction Method of Multipliers (ADMM) [1,[22][23][24][25][26][27][28][29], and methods based on gradient tracking [30][31][32][33][34][35]. Gradient methods typically achieve global attractiveness.…”

Section: Related Work and State Of The Artmentioning

confidence: 99%

“…Nevertheless, they require a correct initialization and, hence, they do not provide global attractiveness. The same issue applies to gradient-tracking methods [30][31][32][33][34][35] (which, anyway, are developed for unconstrained problems), and also for the "node-based" formulations of ADMM [23][24][25][26]28]. Instead, the "edge-based" formulations of ADMM (e.g.…”

Section: Related Work and State Of The Artmentioning

confidence: 99%

A Distributed Methodology for Approximate Uniform Global Minimum Sharing

Bin,

Parisini

2020

Preprint

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The paper deals with the distributed minimum sharing problem, in which a network of decision-makers -exchanging information through a communication network -computes the minimum of some local quantities of interest in a distributed and decentralized way. The problem is equivalently cast into a cost-coupled distributed optimization problem, and an adjustable approximate (or sub-optimal) solution is presented which enjoys several properties of crucial importance in applications. In particular, the proposed solution is scalable in that the dimension of the state space does not grow with the size or topology of the communication network. Moreover, a global and uniform (both in the initial time and in the initial condition) asymptotic stability result is provided, as well as an attractiveness property towards a steady state which can be made arbitrarily close to the sought minimum. Exact asymptotic convergence is also recovered at the price, however, of loosing uniformity of the convergence with respect to the initial time.

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A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

Cited by 22 publications

References 18 publications

Triggered Gradient Tracking for Asynchronous Distributed Optimization

Triggered Gradient Tracking for Asynchronous Distributed Optimization

Extremum Seeking Tracking for Derivative-free Distributed Optimization

A Distributed Methodology for Approximate Uniform Global Minimum Sharing

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