arXiv subjects

Victor M. Preciado

Publications and source records attributed to Victor M. Preciado.

At least 37 records · Page 2Linked to original sources

arXiv2020

A Dynamical Approach to Efficient Eigenvalue Estimation in General Multiagent Networks

We propose a method to efficiently estimate the eigenvalues of any arbitrary (potentially weighted and/or directed) network of interacting dynamical agents from dynamical observations. These observations are discrete, temporal measurements about the evolution of the outputs of a subset of agents (potentially one) during a finite time horizon; notably, we do not require knowledge of which agents are contributing to our measurements. We propose an efficient algorithm to exactly recover the (potentially complex) eigenvalues corresponding to network modes that are observable from the output measurements. The length of the sequence of measurements required by our method to generate a full reconstruction of the observable eigenvalue spectrum is, at most, twice the number of agents in the network, but smaller in practice. The proposed technique can be applied to networks of multiagent systems with arbitrary dynamics in both continuous- and discrete-time. Finally, we illustrate our results with numerical simulations.

math.OC↗

arXiv2020

New bounds on the spectral radius of graphs based on the moment problem

Let $\mathcal{G}$ be an undirected graph with adjacency matrix $A$ and spectral radius $ρ$. Let $w_k, ϕ_k$ and $ϕ_k^{(i)}$ be, respectively, the number walks of length $k$, closed walks of length $k$ and closed walks starting and ending at vertex $i$ after $k$ steps. In this paper, we propose a measure-theoretic framework which allows us to relate walks in a graph with its spectral properties. In particular, we show that $w_k, ϕ_k$ and $ϕ_k^{(i)}$ can be interpreted as the moments of three different measures, all of them supported on the spectrum of $A$. Building on this interpretation, we leverage results from the classical moment problem to formulate a hierarchy of new lower and upper bounds on $ρ$, as well as provide alternative proofs to several well-known bounds in the literature.

math.CO↗

arXiv2020

Robust Closed-loop Model Predictive Control via System Level Synthesis

In this paper, we consider the robust closed-loop model predictive control (MPC) of a linear time-variant (LTV) system with norm bounded disturbances and LTV model uncertainty, wherein a series of constrained optimal control problems (OCPs) are solved. Guaranteeing robust feasibility of these OCPs is challenging due to disturbances perturbing the predicted states, and model uncertainty, both of which can render the closed-loop system unstable. As such, a trade-off between the numerical tractability and conservativeness of the solutions is often required. We use the System Level Synthesis (SLS) framework to reformulate these constrained OCPs over closed-loop system responses, and show that this allows us to transparently account for norm bounded additive disturbances and LTV model uncertainty by computing robust state feedback policies. We further show that by exploiting the underlying linear fractional structure of the resulting robust OCPs, we can significantly reduce the conservativeness of existing SLS-based and tube-MPC-based robust control methods while also improving computational efficiency. We conclude with numerical examples demonstrating the effectiveness of our methods.

math.OC↗

arXiv2020

Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees

The fragility of deep neural networks to adversarially-chosen inputs has motivated the need to revisit deep learning algorithms. Including adversarial examples during training is a popular defense mechanism against adversarial attacks. This mechanism can be formulated as a min-max optimization problem, where the adversary seeks to maximize the loss function using an iterative first-order algorithm while the learner attempts to minimize it. However, finding adversarial examples in this way causes excessive computational overhead during training. By interpreting the min-max problem as an optimal control problem, it has recently been shown that one can exploit the compositional structure of neural networks in the optimization problem to improve the training time significantly. In this paper, we provide the first convergence analysis of this adversarial training algorithm by combining techniques from robust optimal control and inexact oracle methods in optimization. Our analysis sheds light on how the hyperparameters of the algorithm affect the its stability and convergence. We support our insights with experiments on a robust classification problem.

math.OC↗

arXiv2020

Analysis of the susceptible-infected-susceptible epidemic dynamics in networks via the non-backtracking matrix

We study the stochastic susceptible-infected-susceptible model of epidemic processes on finite directed and weighted networks with arbitrary structure. We present a new lower bound on the exponential rate at which the probabilities of nodes being infected decay over time. This bound is directly related to the leading eigenvalue of a matrix that depends on the non-backtracking and incidence matrices of the network. The dimension of this matrix is N+M, where N and M are the number of nodes and edges, respectively. We show that this new lower bound improves on an existing bound corresponding to the so-called quenched mean-field theory. Although the bound obtained from a recently developed second-order moment-closure technique requires the computation of the leading eigenvalue of an N^2 x N^2 matrix, we illustrate in our numerical simulations that the new bound is tighter, while being computationally less expensive for sparse networks. We also present the expression for the corresponding epidemic threshold in terms of the adjacency matrix of the line graph and the non-backtracking matrix of the given network.

physics.soc-ph↗

arXiv2020

Network Design for Controllability Metrics

In this paper, we consider the problem of tuning the edge weights of a networked system described by linear time-invariant dynamics. We assume that the topology of the underlying network is fixed and that the set of feasible edge weights is a given polytope. In this setting, we first consider a feasibility problem consisting of tuning the edge weights such that certain controllability properties are satisfied. The particular controllability properties under consideration are (i) a lower bound on the smallest eigenvalue of the controllability Gramian, which is related to the worst-case energy needed to control the system, and (ii) an upper bound on the trace of the Gramian inverse, which is related to the average control energy. In both cases, the edge-tuning problem can be stated as a feasibility problem involving bilinear matrix equalities, which we approach using a sequence of convex relaxations. Furthermore, we also address a design problem consisting of finding edge weights able to satisfy the aforementioned controllability constraints while seeking to minimize a cost function of the edge weights, which we assume to be convex. In particular, we consider a sparsity-promoting cost function aiming to penalize the number of edges whose weights are modified. Finally, we verify our results with numerical simulations over many random network realizations as well as with an IEEE 14-bus power system topology.

math.OC↗

arXiv2019

Sparse estimation of Laplacian eigenvalues in multiagent networks

We propose a method to efficiently estimate the Laplacian eigenvalues of an arbitrary, unknown network of interacting dynamical agents. The inputs to our estimation algorithm are measurements about the evolution of a collection of agents (potentially one) during a finite time horizon; notably, we do not require knowledge of which agents are contributing to our measurements. We propose a scalable algorithm to exactly recover a subset of the Laplacian eigenvalues from these measurements. These eigenvalues correspond directly to those Laplacian modes that are observable from our measurements. We show how our technique can be applied to networks of multiagent systems with arbitrary dynamics in both continuous- and discrete-time. Finally, we illustrate our results with numerical simulations.

math.SP↗

arXiv2019

Robust Convergence Analysis of Three-Operator Splitting

Operator splitting methods solve composite optimization problems by breaking them into smaller sub-problems that can be solved sequentially or in parallel. In this paper, we propose a unified framework for certifying both linear and sublinear convergence rates for three-operator splitting (TOS) method under a variety of assumptions about the objective function. By viewing the algorithm as a dynamical system with feedback uncertainty (the oracle model), we leverage robust control theory to analyze the worst-case performance of the algorithm using matrix inequalities. We then show how these matrix inequalities can be used to verify sublinear/linear convergence of the TOS algorithm and guide the search for selecting the parameters of the algorithm (both symbolically and numerically) for optimal worst-case performance. We illustrate our results numerically by solving an input-constrained optimal control problem.

math.OC↗

arXiv2019

Contact-Aware Controller Design for Complementarity Systems

While many robotic tasks, like manipulation and locomotion, are fundamentally based in making and breaking contact with the environment, state-of-the-art control policies struggle to deal with the hybrid nature of multi-contact motion. Such controllers often rely heavily upon heuristics or, due to the combinatoric structure in the dynamics, are unsuitable for real-time control. Principled deployment of tactile sensors offers a promising mechanism for stable and robust control, but modern approaches often use this data in an ad hoc manner, for instance to guide guarded moves. In this work, by exploiting the complementarity structure of contact dynamics, we propose a control framework which can close the loop on rich, tactile sensors. Critically, this framework is non-combinatoric, enabling optimization algorithms to automatically synthesize provably stable control policies. We demonstrate this approach on three different underactuated, multi-contact robotics problems.

cs.RO↗

arXiv2019

A Control-Theoretic Approach to Analysis and Parameter Selection of Douglas-Rachford Splitting

Douglas-Rachford splitting and its equivalent dual formulation ADMM are widely used iterative methods in composite optimization problems arising in control and machine learning applications. The performance of these algorithms depends on the choice of step size parameters, for which the optimal values are known in some specific cases, and otherwise are set heuristically. We provide a new unified method of convergence analysis and parameter selection by interpreting the algorithm as a linear dynamical system with nonlinear feedback. This approach allows us to derive a dimensionally independent matrix inequality whose feasibility is sufficient for the algorithm to converge at a specified rate. By analyzing this inequality, we are able to give performance guarantees and parameter settings of the algorithm under a variety of assumptions regarding the convexity and smoothness of the objective function. In particular, our framework enables us to obtain a new and simple proof of the O(1/k) convergence rate of the algorithm when the objective function is not strongly convex.

math.OC↗

arXiv2019

Resilient Structural Stabilizability of Undirected Networks

In this paper, we consider the structural stabilizability problem of undirected networks. More specifically, we are tasked to infer the stabilizability of an undirected network from its underlying topology, where the undirected networks are modeled as continuous-time linear time-invariant (LTI) systems involving symmetric state matrices. Firstly, we derive a graph-theoretic necessary and sufficient condition for structural stabilizability of undirected networks. Then, we propose a method to infer the maximum dimension of stabilizable subspace solely based on the network structure. Based on these results, on one hand, we study the optimal actuator-disabling attack problem, i.e., removing a limited number of actuators to minimize the maximum dimension of stabilizable subspace. We show this problem is NP-hard. On the other hand, we study the optimal recovery problem with respect to the same kind of attacks, i.e., adding a limited number of new actuators such that the maximum dimension of stabilizable subspace is maximized. We prove the optimal recovery problem is also NP-hard, and we develop a (1-1/e) approximation algorithm to this problem.

math.OC↗

arXiv2019

SPECTRE: Seedless Network Alignment via Spectral Centralities

Network alignment consists of finding a structure-preserving correspondence between the nodes of two correlated, but not necessarily identical, networks. This problem finds applications in a wide variety of fields, from the alignment of proteins in computational biology, to the de-anonymization of social networks, as well as recognition tasks in computer vision. In this work we introduce SPECTRE, a scalable algorithm that uses spectral centrality measures and percolation techniques. Unlike most network alignment algorithms, SPECTRE requires no seeds (i.e., pairs of nodes identified beforehand), which in many cases are expensive, or impossible, to obtain. Instead, SPECTRE generates an initial noisy seed set via spectral centrality measures which is then used to robustly grow a network alignment via bootstrap percolation techniques. We show that, while this seed set may contain a majority of incorrect pairs, SPECTRE is still able to obtain a high-quality alignment. Through extensive numerical simulations, we show that SPECTRE allows for fast run times and high accuracy on large synthetic and real-world networks, even those which do not exhibit a high correlation.

cs.SI↗

arXiv2019

Safety Verification of Nonlinear Autonomous System via Occupation Measures

In this paper, we introduce a flexible notion of safety verification for nonlinear autonomous systems by measuring how much time the system spends in given unsafe regions. We consider this problem in the particular case of nonlinear systems with a polynomial dynamics and unsafe regions described by a collection of polynomial inequalities. In this context, we can quantify the amount of time spent in the unsafe regions as the solution to an infinite-dimensional linear program (LP). This LP measures the volume of the unsafe region with respect to the occupation measure of the system trajectories. Using Lasserre hierarchy, we approximate the solution to the infinite-dimensional LP using a sequence of finite-dimensional semidefinite programs (SDPs). The solutions to the SDPs in this hierarchy provide monotonically converging upper bounds on the optimal solution to the infinite-dimensional LP. Finally, we validate the performance of our framework using numerical simulations.

math.OC↗

arXiv2019

Efficient Containment of Exact SIR Markovian Processes on Networks

This paper introduces a theoretical framework for the analysis and control of the stochastic susceptible-infected-removed (SIR) spreading process over a network of heterogeneous agents. In our analysis, we analyze the exact networked Markov process describing the SIR model, without resorting to mean-field approximations, and introduce a convex optimization framework to find an efficient allocation of resources to contain the expected number of accumulated infections over time. Numerical simulations are presented to illustrate the effectiveness of the obtained results.

cs.SI↗

arXiv2019

Stability of SIS Spreading Processes in Networks with Non-Markovian Transmission and Recovery

Although viral spreading processes taking place in networks are often analyzed using Markovian models in which both the transmission and the recovery times follow exponential distributions, empirical studies show that, in many real scenarios, the distribution of these times are not necessarily exponential. To overcome this limitation, we first introduce a generalized susceptible-infected-susceptible (SIS) spreading model that allows transmission and recovery times to follow phase-type distributions. In this context, we derive a lower bound on the exponential decay rate towards the infection-free equilibrium of the spreading model without relying on mean-field approximations. Based on our results, we illustrate how the particular shape of the transmission/recovery distribution influences the exponential rate of convergence towards the equilibrium.

cs.SI↗

arXiv2018

A Chebyshev-Accelerated Primal-Dual Method for Distributed Optimization

We consider a distributed optimization problem over a network of agents aiming to minimize a global objective function that is the sum of local convex and composite cost functions. To this end, we propose a distributed Chebyshev-accelerated primal-dual algorithm to achieve faster ergodic convergence rates. In standard distributed primal-dual algorithms, the speed of convergence towards a global optimum (i.e., a saddle point in the corresponding Lagrangian function) is directly influenced by the eigenvalues of the Laplacian matrix representing the communication graph. In this paper, we use Chebyshev matrix polynomials to generate gossip matrices whose spectral properties result in faster convergence speeds, while allowing for a fully distributed implementation. As a result, the proposed algorithm requires fewer gradient updates at the cost of additional rounds of communications between agents. We illustrate the performance of the proposed algorithm in a distributed signal recovery problem. Our simulations show how the use of Chebyshev matrix polynomials can be used to improve the convergence speed of a primal-dual algorithm over communication networks, especially in networks with poor spectral properties, by trading local computation by communication rounds.

math.OC↗

arXiv2018

Design of First-Order Optimization Algorithms via Sum-of-Squares Programming

In this paper, we propose a framework based on sum-of-squares programming to design iterative first-order optimization algorithms for smooth and strongly convex problems. Our starting point is to develop a polynomial matrix inequality as a sufficient condition for exponential convergence of the algorithm. The entries of this matrix are polynomial functions of the unknown parameters (exponential decay rate, stepsize, momentum coefficient, etc.). We then formulate a polynomial optimization, in which the objective is to optimize the exponential decay rate over the parameters of the algorithm. Finally, we use sum-of-squares programming as a tractable relaxation of the proposed polynomial optimization problem. We illustrate the utility of the proposed framework by designing a first-order algorithm that shares the same structure as Nesterov's accelerated gradient method.

math.OC↗

arXiv2018

Structural Target Controllability of Undirected Networks

In this paper, we study the target controllability problem of networked dynamical systems, in which we are tasked to steer a subset of network states towards a desired objective. More specifically, we derive necessary and sufficient conditions for the structural target controllability problem of linear time-invariant (LTI) systems with symmetric state matrices, such as undirected dynamical networks with unknown link weights. To achieve our goal, we first characterize the generic rank of symmetrically structured matrices, as well as the modes of any numerical realization. Subsequently, we provide a graph-theoretic necessary and sufficient condition for the structural controllability of undirected networks with multiple control nodes. Finally, we derive a graph-theoretic necessary and sufficient condition for structural target controllability of undirected networks. Remarkably, apart from the standard reachability condition, only local topological information is needed for the verification of structural target controllability.

math.OC↗