arXiv subjects

Julien M. Hendrickx

Publications and source records attributed to Julien M. Hendrickx.

At least 55 records · Page 3Linked to original sources

arXiv2022

Automated Performance Estimation for Decentralized Optimization via Network Size Independent Problems

We develop a novel formulation of the Performance Estimation Problem (PEP) for decentralized optimization whose size is independent of the number of agents in the network. The PEP approach allows computing automatically the worst-case performance and worst-case instance of first-order optimization methods by solving an SDP. Unlike previous work, the size of our new PEP formulation is independent of the network size. For this purpose, we take a global view of the decentralized problem and we also decouple the consensus subspace and its orthogonal complement. We apply our methodology to different decentralized methods such as DGD, DIGing and EXTRA and obtain numerically tight performance guarantees that are valid for any network size.

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arXiv2022

Resource allocation in open multi-agent systems: an online optimization analysis

The resource allocation problem consists of the optimal distribution of a budget between agents in a group. We consider such a problem in the context of open systems, where agents can be replaced at some time instances. These replacements lead to variations in both the budget and the total cost function that hinder the overall network's performance. For a simple setting, we analyze the performance of the Random Coordinate Descent algorithm (RCD) using tools similar to those commonly used in online optimization. In particular, we study the accumulated errors that compare solutions issued from the RCD algorithm and the optimal solution or the non-collaborating selfish strategy and we derive some bounds in expectation for these accumulated errors.

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arXiv2022

Minimax Multi-Agent Persistent Monitoring of a Network System

We investigate the problem of optimally observing a finite set of targets using a mobile agent over an infinite time horizon. The agent is tasked to move in a network-constrained structure to gather information so as to minimize the worst-case uncertainty about the internal states of the targets. To do this, the agent has to decide its sequence of target-visits and the corresponding dwell-times at each visited target. For a given visiting sequence, we prove that in an optimal dwelling time allocation the peak uncertainty is the same among all the targets. This allows us to formulate the optimization of dwelling times as a resource allocation problem and to solve it using a novel efficient algorithm. Next, we optimize the visiting sequence using a greedy exploration process, using heuristics inspired by others developed in the context of the traveling salesman problem. Numerical results are included to illustrate the contributions.

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arXiv2022

Fundamental Performance Limitations for Average Consensus in Open Multi-Agent Systems

We derive fundamental performance limitations for intrinsic average consensus problems in open multi-agent systems, which are systems subject to frequent arrivals and departures of agents. Each agent holds a value, and the objective of the agents is to collaboratively estimate the average of the values of the agents presently in the system. Algorithms solving such problems in open systems are poised to never converge because of the permanent variations in the composition, size and objective pursued by the agents of the system. We provide lower bounds on the expected Mean Squared Error achievable by any averaging algorithms in open systems of fixed size. Our derivation is based on the analysis of a conceptual algorithm that would achieve optimal performance for a given model of replacements. We obtain a general bound that depends on the properties of the model defining the interactions between the agents, and instantiate that result for all-to-one and one-to-one interaction models. A comparison between those bounds and algorithms implementable with those models is then provided to highlight their validity.

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arXiv2021

Automated Worst-Case Performance Analysis of Decentralized Gradient Descent

We develop a methodology to automatically compute worst-case performance bounds for a class of decentralized algorithms that optimize the average of local functions distributed across a network. We extend the recently proposed PEP approach to decentralized optimization. This approach allows computing the exact worst-case performance and worst-case instance of centralized algorithms by solving an SDP. We obtain an exact formulation when the network matrix is given, and a relaxation when considering entire classes of network matrices characterized by their spectral range. We apply our methodology to the decentralized (sub)gradient method, obtain a nearly tight worst-case performance bound that significantly improves over the literature, and gain insights into the worst communication networks for a given spectral range.

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arXiv2021

Path-Based Conditions for Local Network Identifiability -- Full Version

This work focuses on the generic identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by transfer functions according to a known topology, some nodes are excited, some are measured, and only a part of the transfer functions are known. Our goal is to determine whether the unknown transfer functions can be generically recovered based on the input-output data collected from the excited and measured nodes. We propose a decoupled version of generic identifiability that is necessary for generic local identifiability and might be equivalent as no counter-example to sufficiency has been found yet in systematic trials. This new notion can be interpreted as the generic identifiability of a larger network, obtained by duplicating the graph, exciting one copy and measuring the other copy. We establish a necessary condition for decoupled identifiability in terms of vertex-disjoint paths in the larger graph, and a sufficient one.

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arXiv2021

Random coordinate descent algorithm for open multi-agent systems with complete topology and homogeneous agents

We study the convergence in expectation of the Random Coordinate Descent algorithm (RCD) for solving optimal resource allocations problems in open multi-agent systems, i.e., multi-agent systems that are subject to arrivals and departures of agents. Assuming all local functions are strongly-convex and smooth, and their minimizers lie in a given ball, we analyse the evolution of the distance to the minimizer in expectation when the system is occasionally subject to replacements in addition to the usual iterations of the RCD algorithm. We focus on complete graphs where all agents interact with each other with the same probability, and provide conditions to guarantee convergence in open system. Finally, a discussion around the tightness of our results is provided.

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arXiv2021

A Semidefinite Programming Approach to Discrete-time Infinite Horizon Persistent Monitoring

We investigate the problem of persistent monitoring, where a mobile agent has to survey multiple targets in an environment in order to estimate their internal states. These internal states evolve with linear stochastic dynamics and the agent can observe them with a linear observation model. However, the signal to noise ratio is a monotonically decreasing function of the distance between the agent and the target. The goal is to minimize the uncertainty in the state estimates over the infinite horizon. We show that, for a periodic trajectory with fixed cycle length, the problem can be formulated as a set of semidefinite programs. We design a scheme that leverages the spatial configuration of the targets to guide the search over this set of optimization problems in order to provide efficient trajectories. Results are compared to a state of the art approach and we obtain improvements of up to 91% in terms of cost in a simple scenario, with much lower computational time.

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arXiv2021

String stability of energy-saving aircraft formations

Groups of aircraft have the potential to save significant amounts of energy by flying in formations; all but the leading aircraft can benefit from the upwash of the wakes of preceding aircraft. A potential obstacle as the number of aircraft in such a formation increases is that disturbances at one aircraft, for example caused by turbulence or wake meandering, can propagate and grow as each following aircraft tries to track the optimal energy-saving position relative to the one in front. This phenomenon, known as string instability, has not yet been adequately examined in the context of aircraft formations. We discuss some trade-offs involved in designing string stable controllers whose objective is to minimize energy, and present a control design method to achieve both string stability and energy efficiency of an aircraft formation. In simulations of a 10-aircraft linear formation in the presence of 2% turbulence intensity, our controller achieves string stability while reduced energy consumption by an average of 13% with respect to solo flight.

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arXiv2021

Are energy savings the only reason for the emergence of bird echelon formation?

We analyze the conditions under which the emergence of frequently observed echelon formation can be explained solely by the maximization of energy savings. We consider a two-dimensional multi-agent echelon formation, where each agent receives a benefit that depends on its position relative to the others, and adjusts its position to increase this benefit. We analyze the selfish case where each agent maximizes its own benefit, leading to a Nash-equilibrium problem, and the collaborative case in which agents maximize the global benefit of the group. We provide conditions on the benefit function under which the frequently observed echelon formations cannot be Nash equilbriums or group optimums. We then show that these conditions are satisfied by the conventionally used fixed-wing wake benefit model. This implies that energy saving alone is not sufficient to explain the emergence of the migratory formations observed, based on the fixed-wing model. Hence, either non-aerodynamic aspects or a more accurate model of bird dynamics should be considered to construct such formations.

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arXiv2020

Local Network Identifiability with Partial Excitation and Measurement

This work focuses on the identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by unknown transfer functions according to a known topology, some nodes are subject to external excitation, and some nodes are measured. The goal is to determine which transfer functions in the network can be recovered based on the input-output data collected from the excited and measured nodes. We propose a local version of network identifiability, representing the ability to recover transfer functions which are approximately known, or to recover them up to a discrete ambiguity. We show that local identifiability is a generic property, establish a necessary and sufficient condition in terms of matrix generic ranks, and exploit this condition to develop an algorithm determining, with probability 1, which transfer functions are locally identifiable. Our implementation presents the results graphically, and is publicly available.

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arXiv2020

Optimal Minimax Mobile Sensor Scheduling Over a Network

We investigate the problem of monitoring multiple targets using a single mobile sensor, with the goal of minimizing the maximum estimation error among all the targets over long time horizons. The sensor can move in a network-constrained structure, where it has to plan which targets to visit and for how long to dwell at each node. We prove that in an optimal observation time allocation, the peak uncertainty is the same among all the targets. By further restricting the agent policy to only visit each target once every cycle, we develop a scheme to optimize the agent's behavior that is significantly simpler computationally when compared to previous approaches for similar problems.

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arXiv2020

Stability of Decentralized Gradient Descent in Open Multi-Agent Systems

The aim of decentralized gradient descent (DGD) is to minimize a sum of $n$ functions held by interconnected agents. We study the stability of DGD in open contexts where agents can join or leave the system, resulting each time in the addition or the removal of their function from the global objective. Assuming all functions are smooth, strongly convex, and their minimizers all lie in a given ball, we characterize the sensitivity of the global minimizer of the sum of these functions to the removal or addition of a new function and provide bounds in $ O\left(\min \left(κ^{0.5}, κ/n^{0.5},κ^{1.5}/n\right)\right)$ where $κ$ is the condition number. We also show that the states of all agents can be eventually bounded independently of the sequence of arrivals and departures. The magnitude of the bound scales with the importance of the interconnection, which also determines the accuracy of the final solution in the absence of arrival and departure, exposing thus a potential trade-off between accuracy and sensitivity. Our analysis relies on the formulation of DGD as gradient descent on an auxiliary function. The tightness of our results is analyzed using the PESTO Toolbox.

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arXiv2020

Low-rank multi-parametric covariance identification

We propose a differential geometric construction for families of low-rank covariance matrices, via interpolation on low-rank matrix manifolds. In contrast with standard parametric covariance classes, these families offer significant flexibility for problem-specific tailoring via the choice of "anchor" matrices for the interpolation. Moreover, their low-rank facilitates computational tractability in high dimensions and with limited data. We employ these covariance families for both interpolation and identification, where the latter problem comprises selecting the most representative member of the covariance family given a data set. In this setting, standard procedures such as maximum likelihood estimation are nontrivial because the covariance family is rank-deficient; we resolve this issue by casting the identification problem as distance minimization. We demonstrate the power of these differential geometric families for interpolation and identification in a practical application: wind field covariance approximation for unmanned aerial vehicle navigation.

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arXiv2020

Multi-Agent Persistent Monitoring of Targets with Uncertain States

We address the problem of persistent monitoring, where a finite set of mobile agents has to persistently visit a finite set of targets. Each of these targets has an internal state that evolves with linear stochastic dynamics. The agents can observe these states, and the observation quality is a function of the distance between the agent and a given target. The goal is then to minimize the mean squared estimation error of these target states. We approach the problem from an infinite horizon perspective, where we prove that, under some natural assumptions, the covariance matrix of each target converges to a limit cycle. The goal, therefore, becomes to minimize the steady state uncertainty. Assuming that the trajectory is parameterized, we provide tools for computing the steady state cost gradient. We show that, in one-dimensional (1D) environments with bounded control and non-overlapping targets, when an optimal control exists it can be represented using a finite number of parameters. We also propose an efficient parameterization of the agent trajectories for multidimensional settings using Fourier curves. Simulation results show the efficacy of the proposed technique in 1D, 2D and 3D scenarios.

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arXiv2020

Exact worst-case convergence rates of the proximal gradient method for composite convex minimization

We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. The proof methodology relies on recent developments in performance estimation of first-order methods based on semidefinite programming. In the case of the proximal gradient method, this methodology allows obtaining exact and non-asymptotic worst-case guarantees that are conceptually very simple, although apparently new. On the way, we discuss how strong convexity can be replaced by weaker assumptions, while preserving the corresponding convergence rates. We also establish that the same fixed step size policy is optimal for all three performance measures. Finally, we extend recent results on the worst-case behavior of gradient descent with exact line search to the proximal case.

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arXiv2019

Exact Worst-case Performance of First-order Methods for Composite Convex Optimization

We provide a framework for computing the exact worst-case performance of any algorithm belonging to a broad class of oracle-based first-order methods for composite convex optimization, including those performing explicit, projected, proximal, conditional and inexact (sub)gradient steps. We simultaneously obtain tight worst-case guarantees and explicit instances of optimization problems on which the algorithm reaches this worst-case. We achieve this by reducing the computation of the worst-case to solving a convex semidefinite program, generalizing previous works on performance estimation by Drori and Teboulle [13] and the authors [43]. We use these developments to obtain a tighter analysis of the proximal point algorithm and of several variants of fast proximal gradient, conditional gradient, subgradient and alternating projection methods. In particular, we present a new analytical worst-case guarantee for the proximal point algorithm that is twice better than previously known, and improve the standard worst-case guarantee for the conditional gradient method by more than a factor of two. We also show how the optimized gradient method proposed by Kim and Fessler in [22] can be extended by incorporating a projection or a proximal operator, which leads to an algorithm that converges in the worst-case twice as fast as the standard accelerated proximal gradient method [2].

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arXiv2019

Optimal Periodic Multi-Agent Persistent Monitoring of a Finite Set of Targets with Uncertain States

We investigate the problem of persistently monitoring a finite set of targets with internal states that evolve with linear stochastic dynamics using a finite set of mobile agents. We approach the problem from the infinite-horizon perspective, looking for periodic movement schedules for the agents. Under linear dynamics and some standard assumptions on the noise distribution, the optimal estimator is a Kalman-Bucy filter and the mean estimation error is a function of its covariance matrix, which evolves as a differential Riccati equation. It is shown that when the agents are constrained to move only over a line and they can see at most one target at a time, the movement policy that minimizes the mean estimation error over time is such that the agent is always either moving with maximum speed or dwelling at a fixed position. This type of trajectory can be fully defined by a finite set of parameters. For periodic trajectories, under some observability conditions, the estimation error converges to a steady state condition and the stochastic gradient estimate of the cost with respect to the trajectory parameters of each agent and the global period can be explicitly computed using Infinitesimal Perturbation Analysis. A gradient-descent approach is used to compute locally optimal parameters. This approach allows us to deal with a very long persistent monitoring horizon using a small number of parameters.

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