arXiv subjects

Shreyas Sundaram

Publications and source records attributed to Shreyas Sundaram.

At least 73 records · Page 4Linked to original sources

arXiv2020

Distributed Hypothesis Testing and Social Learning in Finite Time with a Finite Amount of Communication

We consider the problem of distributed hypothesis testing (or social learning) where a network of agents seeks to identify the true state of the world from a finite set of hypotheses, based on a series of stochastic signals that each agent receives. Prior work on this problem has provided distributed algorithms that guarantee asymptotic learning of the true state, with corresponding efforts to improve the rate of learning. In this paper, we first argue that one can readily modify existing asymptotic learning algorithms to enable learning in finite time, effectively yielding arbitrarily large (asymptotic) rates. We then provide a simple algorithm for finite-time learning which only requires the agents to exchange a binary vector (of length equal to the number of possible hypotheses) with their neighbors at each time-step. Finally, we show that if the agents know the diameter of the network, our algorithm can be further modified to allow all agents to learn the true state and stop transmitting to their neighbors after a finite number of time-steps.

cs.MA↗

arXiv2020

On the Set of Possible Minimizers of a Sum of Known and Unknown Functions

The problem of finding the minimizer of a sum of convex functions is central to the field of optimization. Thus, it is of interest to understand how that minimizer is related to the properties of the individual functions in the sum. In this paper, we consider the scenario where one of the individual functions in the sum is not known completely. Instead, only a region containing the minimizer of the unknown function is known, along with some general characteristics (such as strong convexity parameters). Given this limited information about a portion of the overall function, we provide a necessary condition which can be used to construct an upper bound on the region containing the minimizer of the sum of known and unknown functions. We provide this necessary condition in both the general case where the uncertainty region of the minimizer of the unknown function is arbitrary, and in the specific case where the uncertainty region is a ball.

math.OC↗

arXiv2020

Distributed State Estimation over Time-Varying Graphs: Exploiting the Age-of-Information

We study the problem of designing a distributed observer for an LTI system over a time-varying communication graph. The limited existing work on this topic imposes various restrictions either on the observation model or on the sequence of communication graphs. In contrast, we propose a single-time-scale distributed observer that works under mild assumptions. Specifically, our communication model only requires strong-connectivity to be preserved over non-overlapping, contiguous intervals that are even allowed to grow unbounded over time. We show that under suitable conditions that bound the growth of such intervals, joint observability is sufficient to track the state of any discrete-time LTI system exponentially fast, at any desired rate. In fact, we also establish finite-time convergence based on our approach. Finally, we develop a variant of our algorithm that is provably robust to worst-case adversarial attacks, provided the sequence of graphs is sufficiently connected over time. The key to our approach is the notion of a "freshness-index" that keeps track of the age-of-information being diffused across the network. Such indices enable nodes to reject stale estimates of the state, and, in turn, contribute to stability of the error dynamics.

eess.SY↗

arXiv2020

On the Impacts of Redundancy, Diversity, and Trust in Resilient Distributed State Estimation

We address the problem of distributed state estimation of a linear dynamical process in an attack-prone environment. Recent attempts to solve this problem impose stringent redundancy requirements on the measurement and communication resources of the network. In this paper, we take a step towards alleviating such strict requirements by exploring two complementary directions: (i) making a small subset of the nodes immune to attacks, or "trusted", and (ii) incorporating diversity into the network. We define graph-theoretic constructs that formally capture the notions of redundancy, diversity, and trust. Based on these constructs, we develop a resilient estimation algorithm and demonstrate that even relatively sparse networks that either exhibit node-diversity, or contain a small subset of trusted nodes, can be just as resilient to adversarial attacks as more dense networks. Finally, given a finite budget for network design, we focus on characterizing the complexity of (i) selecting a set of trusted nodes, and (ii) allocating diversity, so as to achieve a desired level of robustness. We establish that, unfortunately, each of these problems is NP-complete.

eess.SY↗

arXiv2019

Resilient Cyberphysical Systems and their Application Drivers: A Technology Roadmap

Cyberphysical systems (CPS) are ubiquitous in our personal and professional lives, and they promise to dramatically improve micro-communities (e.g., urban farms, hospitals), macro-communities (e.g., cities and metropolises), urban structures (e.g., smart homes and cars), and living structures (e.g., human bodies, synthetic genomes). The question that we address in this article pertains to designing these CPS systems to be resilient-from-the-ground-up, and through progressive learning, resilient-by-reaction. An optimally designed system is resilient to both unique attacks and recurrent attacks, the latter with a lower overhead. Overall, the notion of resilience can be thought of in the light of three main sources of lack of resilience, as follows: exogenous factors, such as natural variations and attack scenarios; mismatch between engineered designs and exogenous factors ranging from DDoS (distributed denial-of-service) attacks or other cybersecurity nightmares, so called "black swan" events, disabling critical services of the municipal electrical grids and other connected infrastructures, data breaches, and network failures; and the fragility of engineered designs themselves encompassing bugs, human-computer interactions (HCI), and the overall complexity of real-world systems. In the paper, our focus is on design and deployment innovations that are broadly applicable across a range of CPS application areas.

cs.CY↗

arXiv2019

A Communication-Efficient Algorithm for Exponentially Fast Non-Bayesian Learning in Networks

We introduce a simple time-triggered protocol to achieve communication-efficient non-Bayesian learning over a network. Specifically, we consider a scenario where a group of agents interact over a graph with the aim of discerning the true state of the world that generates their joint observation profiles. To address this problem, we propose a novel distributed learning rule wherein agents aggregate neighboring beliefs based on a min-protocol, and the inter-communication intervals grow geometrically at a rate $a \geq 1$. Despite such sparse communication, we show that each agent is still able to rule out every false hypothesis exponentially fast with probability $1$, as long as $a$ is finite. For the special case when communication occurs at every time-step, i.e., when $a=1$, we prove that the asymptotic learning rates resulting from our algorithm are network-structure independent, and a strict improvement upon those existing in the literature. In contrast, when $a>1$, our analysis reveals that the asymptotic learning rates vary across agents, and exhibit a non-trivial dependence on the network topology coupled with the relative entropies of the agents' likelihood models. This motivates us to consider the problem of allocating signal structures to agents to maximize appropriate performance metrics. In certain special cases, we show that the eccentricity centrality and the decay centrality of the underlying graph help identify optimal allocations; for more general scenarios, we bound the deviation from the optimal allocation as a function of the parameter $a$, and the diameter of the communication graph.

eess.SY↗

arXiv2019

A New Approach to Distributed Hypothesis Testing and Non-Bayesian Learning: Improved Learning Rate and Byzantine-Resilience

We study a setting where a group of agents, each receiving partially informative private signals, seek to collaboratively learn the true underlying state of the world (from a finite set of hypotheses) that generates their joint observation profiles. To solve this problem, we propose a distributed learning rule that differs fundamentally from existing approaches, in that it does not employ any form of "belief-averaging". Instead, agents update their beliefs based on a min-rule. Under standard assumptions on the observation model and the network structure, we establish that each agent learns the truth asymptotically almost surely. As our main contribution, we prove that with probability 1, each false hypothesis is ruled out by every agent exponentially fast at a network-independent rate that is strictly larger than existing rates. We then develop a computationally-efficient variant of our learning rule that is provably resilient to agents who do not behave as expected (as represented by a Byzantine adversary model) and deliberately try to spread misinformation.

eess.SY↗

arXiv2019

A New Approach for Distributed Hypothesis Testing with Extensions to Byzantine-Resilience

We study a setting where a group of agents, each receiving partially informative private observations, seek to collaboratively learn the true state (among a set of hypotheses) that explains their joint observation profiles over time. To solve this problem, we propose a distributed learning rule that differs fundamentally from existing approaches, in the sense, that it does not employ any form of "belief-averaging". Specifically, every agent maintains a local belief (on each hypothesis) that is updated in a Bayesian manner without any network influence, and an actual belief that is updated (up to normalization) as the minimum of its own local belief and the actual beliefs of its neighbors. Under minimal requirements on the signal structures of the agents and the underlying communication graph, we establish consistency of the proposed belief update rule, i.e., we show that the actual beliefs of the agents asymptotically concentrate on the true state almost surely. As one of the key benefits of our approach, we show that our learning rule can be extended to scenarios that capture misbehavior on the part of certain agents in the network, modeled via the Byzantine adversary model. In particular, we prove that each non-adversarial agent can asymptotically learn the true state of the world almost surely, under appropriate conditions on the observation model and the network topology.

eess.SY↗

arXiv2019

Multi-Robot Routing for Persistent Monitoring with Latency Constraints

In this paper we study a multi-robot path planning problem for persistent monitoring of an environment. We represent the areas to be monitored as the vertices of a weighted graph. For each vertex, there is a constraint on the maximum time spent by the robots between visits to that vertex, called the latency, and the objective is to find the minimum number of robots that can satisfy these latency constraints. The decision version of this problem is known to be PSPACE-complete. We present a $O(\log ρ)$ approximation algorithm for the problem where $ρ$ is the ratio of the maximum and the minimum latency constraints. We also present an orienteering based heuristic to solve the problem and show through simulations that in most of the cases the heuristic algorithm gives better solutions than the approximation algorithm. We evaluate our algorithms on large problem instances in a patrolling scenario and in a persistent scene reconstruction application. We also compare the algorithms with an existing solver on benchmark instances.

cs.RO↗

arXiv2019

Game-Theoretic Vaccination Against Networked SIS Epidemics and Impacts of Human Decision-Making

We study decentralized protection strategies against Susceptible-Infected-Susceptible (SIS) epidemics on networks. We consider a population game framework where nodes choose whether or not to vaccinate themselves, and the epidemic risk is defined as the infection probability at the endemic state of the epidemic under a degree-based mean-field approximation. Motivated by studies in behavioral economics showing that humans perceive probabilities and risks in a nonlinear fashion, we specifically examine the impacts of such misperceptions on the Nash equilibrium protection strategies. We first establish the existence and uniqueness of a threshold equilibrium where nodes with degrees larger than a certain threshold vaccinate. When the vaccination cost is sufficiently high, we show that behavioral biases cause fewer players to vaccinate, and vice versa. We quantify this effect for a class of networks with power-law degree distributions by proving tight bounds on the ratio of equilibrium thresholds under behavioral and true perceptions of probabilities. We further characterize the socially optimal vaccination policy and investigate the inefficiency of Nash equilibrium.

cs.GT↗

arXiv2018

On the Location of the Minimizer of the Sum of Two Strongly Convex Functions

The problem of finding the minimizer of a sum of convex functions is central to the field of distributed optimization. Thus, it is of interest to understand how that minimizer is related to the properties of the individual functions in the sum. In this paper, we provide an upper bound on the region containing the minimizer of the sum of two strongly convex functions. We consider two scenarios with different constraints on the upper bound of the gradients of the functions. In the first scenario, the gradient constraint is imposed on the location of the potential minimizer, while in the second scenario, the gradient constraint is imposed on a given convex set in which the minimizers of two original functions are embedded. We characterize the boundaries of the regions containing the minimizer in both scenarios.

math.OC↗

arXiv2018

Finite-Time Distributed State Estimation over Time-Varying Graphs: Exploiting the Age-of-Information

We study the problem of collaboratively estimating the state of a discrete-time LTI process by a network of sensor nodes interacting over a time-varying directed communication graph. Existing approaches to this problem either (i) make restrictive assumptions on the dynamical model, or (ii) make restrictive assumptions on the sequence of communication graphs, or (iii) require multiple consensus iterations between consecutive time-steps of the dynamics, or (iv) require higher-dimensional observers. In this paper, we develop a distributed observer that operates on a single time-scale, is of the same dimension as that of the state, and works under mild assumptions of joint observability of the sensing model, and joint strong-connectivity of the sequence of communication graphs. Our approach is based on the notion of a novel "freshness-index" that keeps track of the age-of-information being diffused across the network. In particular, such indices enable nodes to reject stale information regarding the state of the system, and in turn, help achieve stability of the estimation error dynamics. Based on the proposed approach, the estimate of each node can be made to converge to the true state exponentially fast, at any desired convergence rate. In fact, we argue that finite-time convergence can also be achieved through a suitable selection of the observer gains. Our proof of convergence is self-contained, and employs simple arguments from linear system theory and graph theory.

eess.SY↗

arXiv2018

Byzantine-Resilient Distributed Observers for LTI Systems

Consider a linear time-invariant (LTI) dynamical system monitored by a network of sensors, modeled as nodes of an underlying directed communication graph. We study the problem of collaboratively estimating the state of the system when certain nodes are compromised by adversaries. Specifically, we consider a Byzantine adversary model, where a compromised node possesses complete knowledge of the system dynamics and the network, and can deviate arbitrarily from the rules of any prescribed algorithm. We first characterize certain fundamental limitations of any distributed state estimation algorithm in terms of the measurement and communication structure of the nodes. We then develop an attack-resilient, provably correct state estimation algorithm that admits a fully distributed implementation. To characterize feasible network topologies that guarantee success of our proposed technique, we introduce a notion of `strong-robustness' that captures both measurement and communication redundancy. Finally, by drawing connections to bootstrap percolation theory, we argue that given an LTI system and an associated sensor network, the `strong-robustness' property can be checked in polynomial time.

eess.SY↗

arXiv2018

Game-Theoretic Choice of Curing Rates Against Networked SIS Epidemics by Human Decision-Makers

We study networks of human decision-makers who independently decide how to protect themselves against Susceptible-Infected-Susceptible (SIS) epidemics. Motivated by studies in behavioral economics showing that humans perceive probabilities in a nonlinear fashion, we examine the impacts of such misperceptions on the equilibrium protection strategies. In our setting, nodes choose their curing rates to minimize the infection probability under the degree-based mean-field approximation of the SIS epidemic plus the cost of their selected curing rate. We establish the existence of a degree based equilibrium under both true and nonlinear perceptions of infection probabilities (under suitable assumptions). When the per-unit cost of curing rate is sufficiently high, we show that true expectation minimizers choose the curing rate to be zero at the equilibrium, while curing rate is nonzero under nonlinear probability weighting.

cs.GT↗

arXiv2018

A Resilient Convex Combination for consensus-based distributed algorithms

Consider a set of vectors in $\mathbb{R}^n$, partitioned into two classes: normal vectors and malicious vectors. The number of malicious vectors is bounded but their identities are unknown. The paper provides a way for achieving a resilient convex combination, which is a convex combination of only normal vectors. Compared with existing approaches based on Tverberg points, the proposed method based on the intersection of convex hulls has lower computational complexity. Simulations suggest that the proposed method can be applied to resilience for consensus-based distributed algorithms against Byzantine attacks.

math.OC↗

arXiv2018

On the Complexity and Approximability of Optimal Sensor Selection for Kalman Filtering

Given a linear dynamical system, we consider the problem of selecting (at design-time) an optimal set of sensors (subject to certain budget constraints) to minimize the trace of the steady state error covariance matrix of the Kalman filter. Previous work has shown that this problem is NP-hard for certain classes of systems and sensor costs; in this paper, we show that the problem remains NP-hard even for the special case where the system is stable and all sensor costs are identical. Furthermore, we show the stronger result that there is no constant-factor (polynomial-time) approximation algorithm for this problem. This contrasts with other classes of sensor selection problems studied in the literature, which typically pursue constant-factor approximations by leveraging greedy algorithms and submodularity of the cost function. Here, we provide a specific example showing that greedy algorithms can perform arbitrarily poorly for the problem of design-time sensor selection for Kalman filtering.

math.OC↗

arXiv2018

Min-Max Tours for Task Allocation to Heterogeneous Agents

We consider a scenario consisting of a set of heterogeneous mobile agents located at a depot, and a set of tasks dispersed over a geographic area. The agents are partitioned into different types. The tasks are partitioned into specialized tasks that can only be done by agents of a certain type, and generic tasks that can be done by any agent. The distances between each pair of tasks are specified, and satisfy the triangle inequality. Given this scenario, we address the problem of allocating these tasks among the available agents (subject to type compatibility constraints) while minimizing the maximum cost to tour the allocation by any agent and return to the depot. This problem is NP-hard, and we give a three phase algorithm to solve this problem that provides 5-factor approximation, regardless of the total number of agents and the number of agents of each type. We also show that in the special case where there is only one agent of each type, the algorithm has an approximation factor of 4.

cs.MA↗

arXiv2018

Secure Distributed State Estimation of an LTI System over Time-Varying Networks and Analog Erasure Channels

We study the problem of collaboratively estimating the state of an LTI system monitored by a network of sensors, subject to the following important practical considerations: (i) certain sensors might be arbitrarily compromised by an adversary and (ii) the underlying communication graph governing the flow of information across sensors might be time-varying. We first analyze a scenario involving intermittent communication losses that preserve certain information flow patterns over bounded intervals of time. By equipping the sensors with adequate memory, we show that one can obtain a fully distributed, provably correct state estimation algorithm that accounts for arbitrary adversarial behavior, provided certain conditions are met by the network topology. We then argue that our approach can handle bounded communication delays as well. Next, we explore a case where each communication link stochastically drops packets based on an analog erasure channel model. For this setup, we propose state estimate update and information exchange rules, along with conditions on the network topology and packet drop probabilities, that guarantee mean-square stability despite arbitrary adversarial attacks.

eess.SY↗