arXiv subjects

Usman A. Khan

Publications and source records attributed to Usman A. Khan.

At least 55 records · Page 3Linked to original sources

arXiv2018

FROST -- Fast row-stochastic optimization with uncoordinated step-sizes

In this paper, we discuss distributed optimization over directed graphs, where doubly-stochastic weights cannot be constructed. Most of the existing algorithms overcome this issue by applying push-sum consensus, which utilizes column-stochastic weights. The formulation of column-stochastic weights requires each agent to know (at least) its out-degree, which may be impractical in e.g., broadcast-based communication protocols. In contrast, we describe FROST (Fast Row-stochastic-Optimization with uncoordinated STep-sizes), an optimization algorithm applicable to directed graphs that does not require the knowledge of out-degrees; the implementation of which is straightforward as each agent locally assigns weights to the incoming information and locally chooses a suitable step-size. We show that FROST converges linearly to the optimal solution for smooth and strongly-convex functions given that the largest step-size is positive and sufficiently small.

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arXiv2018

Optimization over time-varying directed graphs with row and column-stochastic matrices

In this paper, we provide a distributed optimization algorithm, termed as TV-$\mathcal{AB}$, that minimizes a sum of convex functions over time-varying, random directed graphs. Contrary to the existing work, the algorithm we propose does not require eigenvector estimation to estimate the (non-$\mathbf{1}$) Perron eigenvector of a stochastic matrix. Instead, the proposed approach relies on a novel information mixing approach that exploits both row- and column-stochastic weights to achieve agreement towards the optimal solution when the underlying graph is directed. We show that TV-$\mathcal{AB}$ converges linearly to the optimal solution when the global objective is smooth and strongly-convex, and the underlying time-varying graphs exhibit bounded connectivity, i.e., a union of every $C$ consecutive graphs is strongly-connected. We derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument. Simulations confirm the findings in this paper.

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arXiv2018

Distributed heavy-ball: A generalization and acceleration of first-order methods with gradient tracking

We study distributed optimization to minimize a global objective that is a sum of smooth and strongly-convex local cost functions. Recently, several algorithms over undirected and directed graphs have been proposed that use a gradient tracking method to achieve linear convergence to the global minimizer. However, a connection between these different approaches has been unclear. In this paper, we first show that many of the existing first-order algorithms are in fact related with a simple state transformation, at the heart of which lies the $\mathcal{AB}$ algorithm. We then describe \textit{distributed heavy-ball}, denoted as $\mathcal{AB}m$, i.e., $\mathcal{AB}$ with momentum, that combines gradient tracking with a momentum term and uses nonidentical local step-sizes. By simultaneously implementing both row- and column-stochastic weights, $\mathcal{AB}m$ removes the conservatism in the related work due to doubly-stochastic weights or eigenvector estimation. $\mathcal{AB}m$ thus naturally leads to optimization and average-consensus over both undirected and directed graphs, casting a unifying framework over several well-known consensus algorithms over arbitrary strongly-connected graphs. We show that $\mathcal{AB}m$ has a global $R$-linear rate when the largest step-size is positive and sufficiently small. Following the standard practice in the heavy-ball literature, we numerically show that $\mathcal{AB}m$ achieves accelerated convergence especially when the objective function is ill-conditioned.

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arXiv2018

A linear algorithm for optimization over directed graphs with geometric convergence

In this letter, we study distributed optimization, where a network of agents, abstracted as a directed graph, collaborates to minimize the average of locally-known convex functions. Most of the existing approaches over directed graphs are based on push-sum (type) techniques, which use an independent algorithm to asymptotically learn either the left or right eigenvector of the underlying weight matrices. This strategy causes additional computation, communication, and nonlinearity in the algorithm. In contrast, we propose a linear algorithm based on an inexact gradient method and a gradient estimation technique. Under the assumptions that each local function is strongly-convex with Lipschitz-continuous gradients, we show that the proposed algorithm geometrically converges to the global minimizer with a sufficiently small step-size. We present simulations to illustrate the theoretical findings.

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arXiv2018

Structural cost-optimal design of sensor networks for distributed estimation

In this letter we discuss cost optimization of sensor networks monitoring structurally full-rank systems under distributed observability constraint. Using structured systems theory, the problem is relaxed into two subproblems: (i) sensing cost optimization and (ii) networking cost optimization. Both problems are reformulated as combinatorial optimization problems. The sensing cost optimization is shown to have a polynomial order solution. The networking cost optimization is shown to be NP-hard in general, but has a polynomial order solution under specific conditions. A 2-approximation polynomial order relaxation is provided for general networking cost optimization, which is applicable in large-scale system monitoring.

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arXiv2018

Localization in internets of mobile agents: A linear approach

Fifth generation~(5G) networks providing much higher bandwidth and faster data rates will allow connecting vast number of static and mobile devices, sensors, agents, users, machines, and vehicles, supporting Internet-of-Things (IoT), real-time dynamic networks of mobile things. Positioning and location awareness will become increasingly important, enabling deployment of new services and contributing to significantly improving the overall performance of the 5G~system. Many of the currently talked about solutions to positioning in~5G are centralized, mostly requiring direct line-of-sight (LoS) to deployed access nodes or anchors at the same time, which in turn requires high-density deployments of anchors. But these LoS and centralized positioning solutions may become unwieldy as the number of users and devices continues to grow without limit in sight. As an alternative to the centralized solutions, this paper discusses distributed localization in a 5G enabled IoT environment where many low power devices, users, or agents are to locate themselves without global or LoS access to anchors. Even though positioning is essentially a non-linear problem (solving circle equations by trilateration or triangulation), we discuss a cooperative \textit{linear} distributed iterative solution with only local measurements, communication and computation needed at each agent. Linearity is obtained by reparametrization of the agent location through barycentric coordinate representations based on local neighborhood geometry that may be computed in terms of certain Cayley-Menger determinants involving relative local inter-agent distance measurements.

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arXiv2017

Linear convergence in optimization over directed graphs with row-stochastic matrices

This paper considers a distributed optimization problem over a multi-agent network, in which the objective function is a sum of individual cost functions at the agents. We focus on the case when communication between the agents is described by a \emph{directed} graph. Existing distributed optimization algorithms for directed graphs require at least the knowledge of the neighbors' out-degree at each agent (due to the requirement of column-stochastic matrices). In contrast, our algorithm requires no such knowledge. Moreover, the proposed algorithm achieves the best known rate of convergence for this class of problems, $O(μ^k)$ for $0<μ<1$, where $k$ is the number of iterations, given that the objective functions are strongly-convex and have Lipschitz-continuous gradients. Numerical experiments are also provided to illustrate the theoretical findings.

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arXiv2017

Distributed Estimation Recovery under Sensor Failure

Single time-scale distributed estimation of dynamic systems via a network of sensors/estimators is addressed in this letter. In single time-scale distributed estimation, the two fusion steps, consensus and measurement exchange, are implemented only once, in contrast to, e.g., a large number of consensus iterations at every step of the system dynamics. We particularly discuss the problem of failure in the sensor/estimator network and how to recover for distributed estimation by adding new sensor measurements from equivalent states. We separately discuss the recovery for two types of sensors, namely αand βsensors. We propose polynomial order algorithms to find equivalent state nodes in graph representation of system to recover for distributed observability. The polynomial order solution is particularly significant for large-scale systems.

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arXiv2017

ADD-OPT: Accelerated Distributed Directed Optimization

In this paper, we consider distributed optimization problems where the goal is to minimize a sum of objective functions over a multi-agent network. We focus on the case when the inter-agent communication is described by a strongly-connected, \emph{directed} graph. The proposed algorithm, ADD-OPT (Accelerated Distributed Directed Optimization), achieves the best known convergence rate for this class of problems,~$O(μ^{k}),0<μ<1$, given strongly-convex, objective functions with globally Lipschitz-continuous gradients, where~$k$ is the number of iterations. Moreover, ADD-OPT supports a wider and more realistic range of step-sizes in contrast to existing work. In particular, we show that ADD-OPT converges for arbitrarily small (positive) step-sizes. Simulations further illustrate our results.

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arXiv2017

Distributed Subgradient Projection Algorithm over Directed Graphs: Alternate Proof

We propose Directed-Distributed Projected Subgradient (D-DPS) to solve a constrained optimization problem over a multi-agent network, where the goal of agents is to collectively minimize the sum of locally known convex functions. Each agent in the network owns only its local objective function, constrained to a commonly known convex set. We focus on the circumstance when communications between agents are described by a \emph{directed} network. The D-DPS combines surplus consensus to overcome the asymmetry caused by the directed communication network. The analysis shows the convergence rate to be $O(\frac{\ln k}{\sqrt{k}})$.

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arXiv2017

Localization in mobile networks via virtual convex hulls

In this paper, we develop a \textit{distributed} algorithm to localize an arbitrary number of agents moving in a bounded region of interest. We assume that the network contains \textit{at least one} agent with known location (hereinafter referred to as an anchor), and each agent measures a noisy version of its motion and the distances to the nearby agents. We provide a~\emph{geometric approach}, which allows each agent to: (i) continually update the distances to the locations where it has exchanged information with the other nodes in the past; and (ii) measure the distance between a neighbor and any such locations. Based on this approach, we provide a \emph{linear update} to find the locations of an arbitrary number of mobile agents when they follow some convexity in their deployment and motion. Since the agents are mobile, they may not be able to find nearby nodes (agents and/or anchors) to implement a distributed algorithm. To address this issue, we introduce the notion of a \emph{virtual convex hull} with the help of the aforementioned geometric approach. In particular, each agent keeps track of a virtual convex hull of other nodes, which may not physically exist, and updates its location with respect to its neighbors in the virtual hull. We show that the corresponding localization algorithm, in the absence of noise, can be abstracted as a Linear Time-Varying (LTV) system, with non-deterministic system matrices, which asymptotically tracks the true locations of the agents. We provide simulations to verify the analytical results and evaluate the performance of the algorithm in the presence of noise on the motion as well as on the distance measurements.

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arXiv2016

Distributed Subgradient Projection Algorithm over Directed Graphs

We propose a distributed algorithm, termed the Directed-Distributed Projected Subgradient (D-DPS), to solve a constrained optimization problem over a multi-agent network, where the goal of agents is to collectively minimize the sum of locally known convex functions. Each agent in the network owns only its local objective function, constrained to a commonly known convex set. We focus on the circumstance when communications between agents are described by a directed network. The D-DPS augments an additional variable for each agent, to overcome the asymmetry caused by the directed communication network. The convergence analysis shows that D-DPS converges at a rate of $O(\frac{\ln k}{\sqrt{k}})$, where k is the number of iterations.

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arXiv2016

On the Linear Convergence of Distributed Optimization over Directed Graphs

This paper develops a fast distributed algorithm, termed \emph{DEXTRA}, to solve the optimization problem when~$n$ agents reach agreement and collaboratively minimize the sum of their local objective functions over the network, where the communication between the agents is described by a~\emph{directed} graph. Existing algorithms solve the problem restricted to directed graphs with convergence rates of $O(\ln k/\sqrt{k})$ for general convex objective functions and $O(\ln k/k)$ when the objective functions are strongly-convex, where~$k$ is the number of iterations. We show that, with the appropriate step-size, DEXTRA converges at a linear rate $O(τ^{k})$ for $0<τ<1$, given that the objective functions are restricted strongly-convex. The implementation of DEXTRA requires each agent to know its local out-degree. Simulation examples further illustrate our findings.

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arXiv2016

On the Distributed Optimization over Directed Networks

In this paper, we propose a distributed algorithm, called Directed-Distributed Gradient Descent (D-DGD), to solve multi-agent optimization problems over directed graphs. Existing algorithms mostly deal with similar problems under the assumption of undirected networks, i.e., requiring the weight matrices to be doubly-stochastic. The row-stochasticity of the weight matrix guarantees that all agents reach consensus, while the column-stochasticity ensures that each agent's local gradient contributes equally to the global objective. In a directed graph, however, it may not be possible to construct a doubly-stochastic weight matrix in a distributed manner. We overcome this difficulty by augmenting an additional variable for each agent to record the change in the state evolution. In each iteration, the algorithm simultaneously constructs a row-stochastic matrix and a column-stochastic matrix instead of only a doubly-stochastic matrix. The convergence of the new weight matrix, depending on the row-stochastic and column-stochastic matrices, ensures agents to reach both consensus and optimality. The analysis shows that the proposed algorithm converges at a rate of $O(\frac{\ln k}{\sqrt{k}})$, where $k$ is the number of iterations.

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arXiv2015

Distributed Mirror Descent over Directed Graphs

In this paper, we propose Distributed Mirror Descent (DMD) algorithm for constrained convex optimization problems on a (strongly-)connected multi-agent network. We assume that each agent has a private objective function and a constraint set. The proposed DMD algorithm employs a locally designed Bregman distance function at each agent, and thus can be viewed as a generalization of the well-known Distributed Projected Subgradient (DPS) methods, which use identical Euclidean distances at the agents. At each iteration of the DMD, each agent optimizes its own objective adjusted with the Bregman distance function while exchanging state information with its neighbors. To further generalize DMD, we consider the case where the agent communication follows a \emph{directed} graph and it may not be possible to design doubly-stochastic weight matrices. In other words, we restrict the corresponding weight matrices to be row-stochastic instead of doubly-stochastic. We study the convergence of DMD in two cases: (i) when the constraint sets at the agents are the same; and, (ii) when the constraint sets at the agents are different. By partially following the spirit of our proof, it can be shown that a class of consensus-based distributed optimization algorithms, restricted to doubly-stochastic matrices, remain convergent with stochastic matrices.

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arXiv2014

Asymptotic stability of stochastic LTV systems with applications to distributed dynamic fusion

In this paper, we investigate asymptotic stability of linear time-varying systems with (sub-) stochastic system matrices. Motivated by distributed dynamic fusion over networks of mobile agents, we impose some mild regularity conditions on the elements of time-varying system matrices. We provide sufficient conditions under which the asymptotic stability of the LTV system can be guaranteed. By introducing the notion of slices, as non-overlapping partitions of the sequence of systems matrices, we obtain stability conditions in terms of the slice lengths and some network parameters. In addition, we apply the LTV stability results to the distributed leader-follower algorithm, and show the corresponding convergence and steady-state. An illustrative example is also included to validate the effectiveness of our approach.

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arXiv2014

Measurement partitioning and observational equivalence in state estimation

This letter studies measurement partitioning and equivalence in state estimation based on graph-theoretic principles. We show that a set of critical measurements (required to ensure LTI state-space observability) can be further partitioned into two types:~$α$ and~$β$. This partitioning is driven by different graphical (or algebraic) methods used to define the corresponding measurements. Subsequently, we describe observational equivalence, i.e. given an~$α$ (or~$β$) measurement, say~$y_i$, what is the set of measurements equivalent to~$y_i$, such that only one measurement in this set is required to ensure observability? Since~$α$ and~$β$ measurements are cast using different algebraic and graphical characteristics, their equivalence sets are also derived using different algebraic and graph-theoretic principles. We illustrate the related concepts on an appropriate system digraph.

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arXiv2014

Graphic-theoretic distributed inference in social networks

We consider distributed inference in social networks where a phenomenon of interest evolves over a given social interaction graph, referred to as the \emph{social digraph}. For inference, we assume that a network of agents monitors certain nodes in the social digraph and no agent may be able to perform inference within its neighborhood; the agents must rely on inter-agent communication. The key contributions of this paper include: (i) a novel construction of the distributed estimator and distributed observability from the first principles; (ii) a graph-theoretic agent classification that establishes the importance and role of each agent towards inference; (iii) characterizing the necessary conditions, based on the classification in (ii), on the agent network to achieve distributed observability. Our results are based on structured systems theory and are applicable to any parameter choice of the underlying system matrix as long as the social digraph remains fixed. In other words, any social phenomena that evolves (linearly) over a structure-invariant social digraph may be considered--we refer to such systems as Liner Structure-Invariant (LSI). The aforementioned contributions, (i)--(iii), thus, only require the knowledge of the social digraph (topology) and are independent of the social phenomena. We show the applicability of the results to several real-wold social networks, i.e. social influence among monks, networks of political blogs and books, and a co-authorship graph.

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