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Matthew Hale

Publications and source records attributed to Matthew Hale.

72 records · Page 4Linked to original sources
arXiv2020

Differentially Private Formation Control

As multi-agent systems proliferate, there is increasing demand for coordination protocols that protect agents' sensitive information while allowing them to collaborate. To help address this need, this paper presents a differentially private formation control framework. Agents' state trajectories are protected using differential privacy, which is a statistical notion of privacy that protects data by adding noise to it. We provide a private formation control implementation and analyze the impact of privacy upon the system. Specifically, we quantify tradeoffs between privacy level, system performance, and connectedness of the network's communication topology. These tradeoffs are used to develop guidelines for calibrating privacy in terms of control theoretic quantities, such as steady-state error, without requiring in-depth knowledge of differential privacy. Additional guidelines are also developed for treating privacy levels and network topologies as design parameters to tune the network's performance. Simulation results illustrate these tradeoffs and show that strict privacy is inherently compatible with strong system performance.

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arXiv2020

Towards Totally Asynchronous Primal-Dual Convex Optimization in Blocks

We present a parallelized primal-dual algorithm for solving constrained convex optimization problems. The algorithm is "block-based," in that vectors of primal and dual variables are partitioned into blocks, each of which is updated only by a single processor. We consider four possible forms of asynchrony: in updates to primal variables, updates to dual variables, communications of primal variables, and communications of dual variables. We explicitly construct a family of counterexamples to rule out permitting asynchronous communication of dual variables, though the other forms of asynchrony are permitted, all without requiring bounds on delays. A first-order update law is developed and shown to be robust to asynchrony. We then derive convergence rates to a Lagrangian saddle point in terms of the operations agents execute, without specifying any timing or pattern with which they must be executed. These convergence rates contain a synchronous algorithm as a special case and are used to quantify an "asynchrony penalty." Numerical results illustrate these developments.

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arXiv2020

Non-Asymptotic Connectivity of Random Graphs and Their Unions

Graph-theoretic methods have seen wide use throughout the literature on multi-agent control and optimization. When communications are intermittent and unpredictable, such networks have been modeled using random communication graphs. When graphs are time-varying, it is common to assume that their unions are connected over time, yet, to the best of our knowledge, there are not results that determine the number of finite-size random graphs needed to attain a connected union. Therefore, this paper bounds the probability that individual random graphs are connected and bounds the same probability for connectedness of unions of random graphs. The random graph model used is a generalization of the classic Erdos-Renyi model which allows some edges never to appear. Numerical results are presented to illustrate the analytical developments made.

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arXiv2020

Predictive resource allocation for flexible loads with local QoS

Loads that can vary their power consumption without violating their Quality of service (QoS), that is flexible loads, are an invaluable resource for grid operators. Utilizing flexible loads as a resource requires the grid operator to incorporate them into a resource allocation problem. Since flexible loads are often consumers, for concerns of privacy it is desirable for this problem to have a distributed implementation. Technically, this distributed implementation manifests itself as a time varying convex optimization problem constrained by the QoS of each load. In the literature, a time invariant form of this problem without all of the necessary QoS metrics for the flexible loads is often considered. Moving to a more realistic setup introduces additional technical challenges, due to the problems' time-varying nature. In this work, we develop an algorithm to account for the challenges introduced when considering a time varying setup with appropriate QoS metrics.

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arXiv2020

Privacy-Preserving Policy Synthesis in Markov Decision Processes

In decision-making problems, the actions of an agent may reveal sensitive information that drives its decisions. For instance, a corporation's investment decisions may reveal its sensitive knowledge about market dynamics. To prevent this type of information leakage, we introduce a policy synthesis algorithm that protects the privacy of the transition probabilities in a Markov decision process. We use differential privacy as the mathematical definition of privacy. The algorithm first perturbs the transition probabilities using a mechanism that provides differential privacy. Then, based on the privatized transition probabilities, we synthesize a policy using dynamic programming. Our main contribution is to bound the "cost of privacy," i.e., the difference between the expected total rewards with privacy and the expected total rewards without privacy. We also show that computing the cost of privacy has time complexity that is polynomial in the parameters of the problem. Moreover, we establish that the cost of privacy increases with the strength of differential privacy protections, and we quantify this increase. Finally, numerical experiments on two example environments validate the established relationship between the cost of privacy and the strength of data privacy protections.

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arXiv2020

An Algorithm for Multi-Objective Multi-Agent Optimization

Multi-agent optimization problems with many objective functions have drawn much interest over the past two decades. Many works on the subject minimize the sum of objective functions, which implicitly carries a decision about the problem formulation. Indeed, it represents a special case of a multi-objective problem, in which all objectives are prioritized equally. To the best of our knowledge, multi-objective optimization applied to multi-agent systems remains largely unexplored. Therefore, we propose a distributed algorithm that allows the exploration of Pareto optimal solutions for the non-homogeneously weighted sum of objective functions. In the problems we consider, each agent has one objective function to minimize based on a gradient method. Agents update their decision variables by exchanging information with other agents in the network. Information exchanges are weighted by each agent's individual weights that encode the extent to which they prioritize other agents' objectives. This paper provides a proof of convergence, performance bounds, and explicit limits for the results of agents' computations. Simulation results with different sizes of networks demonstrate the efficiency of the proposed approach and how the choice of weights impacts the agents' final result.

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arXiv2019

The Dirichlet Mechanism for Differential Privacy on the Unit Simplex

As members of a network share more information with each other and network providers, sensitive data leakage raises privacy concerns. To address this need for a class of problems, we introduce a novel mechanism that privatizes vectors belonging to the unit simplex. Such vectors can be seen in many applications, such as privatizing a decision-making policy in a Markov decision process. We use differential privacy as the underlying mathematical framework for these developments. The introduced mechanism is a probabilistic mapping that maps a vector within the unit simplex to the same domain according to a Dirichlet distribution. We find the mechanism well-suited for inputs within the unit simplex because it always returns a privatized output that is also in the unit simplex. Therefore, no further projection back onto the unit simplex is required. We verify the privacy guarantees of the mechanism for two cases, namely, identity queries and average queries. In the former case, we derive expressions for the differential privacy level of privatizing a single vector within the unit simplex. In the latter case, we study the mechanism for privatizing the average of a collection of vectors, each of which is in the unit simplex. We establish a trade-off between the strength of privacy and the variance of the mechanism output, and we introduce a parameter to balance the trade-off between them. Numerical results illustrate these developments.

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arXiv2019

Error Bounds and Guidelines for Privacy Calibration in Differentially Private Kalman Filtering

Differential privacy has emerged as a formal framework for protecting sensitive information in control systems. One key feature is that it is immune to post-processing, which means that arbitrary post-hoc computations can be performed on privatized data without weakening differential privacy. It is therefore common to filter private data streams. To characterize this setup, in this paper we present error and entropy bounds for Kalman filtering differentially private state trajectories. We consider systems in which an output trajectory is privatized in order to protect the state trajectory that produced it. We provide bounds on a priori and a posteriori error and differential entropy of a Kalman filter which is processing the privatized output trajectories. Using the error bounds we develop, we then provide guidelines to calibrate privacy levels in order to keep filter error within pre-specified bounds. Simulation results are presented to demonstrate these developments.

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arXiv2019

Bidirectional Information Flow and the Roles of Privacy Masks in Cloud-Based Control

We consider a cloud-based control architecture for a linear plant with Gaussian process noise, where the state of the plant contains a client's sensitive information. We assume that the cloud tries to estimate the state while executing a designated control algorithm. The mutual information between the client's actual state and the cloud's estimate is adopted as a measure of privacy loss. We discuss the necessity of uplink and downlink privacy masks. After observing that privacy is not necessarily a monotone function of the noise levels of privacy masks, we discuss the joint design procedure for uplink and downlink privacy masks. Finally, the trade-off between privacy and control performance is explored.

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arXiv2019

Totally Asynchronous Distributed Quadratic Programming with Independent Stepsizes and Regularizations

Quadratic programs arise in robotics, communications, smart grids, and many other applications. As these problems grow in size, finding solutions becomes much more computationally demanding, and new algorithms are needed to efficiently solve them. Targeting large-scale problems, we develop a multi-agent quadratic programming framework in which each agent updates only a small number of the total decision variables in a problem. Agents communicate their updated values to each other, though we do not impose any restrictions on the timing with which they do so, nor on the delays in these transmissions. Furthermore, we allow weak parametric coupling among agents, in the sense that they are free to independently choose their stepsizes, subject to mild restrictions. We show that these stepsize restrictions depend upon a problem's condition number. We further provide the means for agents to independently regularize the problem they solve, thereby improving condition numbers and, as we will show, convergence properties, while preserving agents' independence in selecting parameters. Simulation results are provided to demonstrate the success of this framework on a practical quadratic program.

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arXiv2019

Privacy in Feedback: The Differentially Private LQG

Information communicated within cyber-physical systems (CPSs) is often used in determining the physical states of such systems, and malicious adversaries may intercept these communications in order to infer future states of a CPS or its components. Accordingly, there arises a need to protect the state values of a system. Recently, the notion of differential privacy has been used to protect state trajectories in dynamical systems, and it is this notion of privacy that we use here to protect the state trajectories of CPSs. We incorporate a cloud computer to coordinate the agents comprising the CPSs of interest, and the cloud offers the ability to remotely coordinate many agents, rapidly perform computations, and broadcast the results, making it a natural fit for systems with many interacting agents or components. Striving for broad applicability, we solve infinite-horizon linear-quadratic-regulator (LQR) problems, and each agent protects its own state trajectory by adding noise to its states before they are sent to the cloud. The cloud then uses these state values to generate optimal inputs for the agents. As a result, private data is fed into feedback loops at each iteration, and each noisy term affects every future state of every agent. In this paper, we show that the differentially private LQR problem can be related to the well-studied linear-quadratic-Gaussian (LQG) problem, and we provide bounds on how agents' privacy requirements affect the cloud's ability to generate optimal feedback control values for the agents. These results are illustrated in numerical simulations.

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arXiv2019

Differentially Private Smart Metering: Implementation, Analytics, and Billing

Smart power grids offer to revolutionize power distribution by sharing granular power usage data, though this same data sharing can reveal a great deal about users, and there are serious privacy concerns for customers. In this paper, we address these concerns using differential privacy. Differential privacy is a statistical notion of privacy that adds carefully-calibrated noise to sensitive data before sharing it, and we apply it to provide strong privacy guarantees to individual customers. Under this implementation, we quantify the impact of privacy upon smart grid analytics, which are of interest to utilities, to quantify the loss in accuracy incurred by privacy. Simultaneously, we quantify the financial impact of privacy through bounding deviations in customers' utility bills. Simulation results are provided using actual power usage data, demonstrating the viability of this approach in practice.

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arXiv2018

Towards Differential Privacy for Symbolic Systems

In this paper, we develop a privacy implementation for symbolic control systems. Such systems generate sequences of non-numerical data, and these sequences can be represented by words or strings over a finite alphabet. This work uses the framework of differential privacy, which is a statistical notion of privacy that makes it unlikely that privatized data will reveal anything meaningful about underlying sensitive data. To bring differential privacy to symbolic control systems, we develop an exponential mechanism that approximates a sensitive word using a randomly chosen word that is likely to be near it. The notion of "near" is given by the Levenshtein distance, which counts the number of operations required to change one string into another. We then develop a Levenshtein automaton implementation of our exponential mechanism that efficiently generates privatized output words. This automaton has letters as its states, and this work develops transition probabilities among these states that give overall output words obeying the distribution required by the exponential mechanism. Numerical results are provided to demonstrate this technique for both strings of English words and runs of a deterministic transition system, demonstrating in both cases that privacy can be provided in this setting while maintaining a reasonable degree of accuracy.

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arXiv2018

Asynchronous Distributed Optimization with Heterogeneous Regularizations and Normalizations

As multi-agent networks grow in size and scale, they become increasingly difficult to synchronize, though agents must work together even when generating and sharing different information at different times. Targeting such cases, this paper presents an asynchronous optimization framework in which the time between successive communications and computations is unknown and unspecified for each agent. Agents' updates are carried out in blocks, with each agent updating only a small subset of all decision variables. To provide robustness to asynchrony, each agent uses an independently chosen Tikhonov regularization. Convergence is measured with respect to a weighted block-maximum norm in which convergence of agents' blocks can be measured in different p-norms and weighted differently to heterogeneously normalize problems. Asymptotic convergence is shown and convergence rates are derived explicitly in terms of a problem's parameters, with only mild restrictions imposed upon them. Simulation results are provided to verify the theoretical developments made.

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arXiv2017

Differentially Private Cloud-Based Multi-Agent Optimization with Constraints

We present an optimization framework that solves constrained multi-agent optimization problems while keeping each agent's state differentially private. The agents in the network seek to optimize a local objective function in the presence of global constraints. Agents communicate only through a trusted cloud computer and the cloud also performs computations based on global information. The cloud computer modifies the results of such computations before they are sent to the agents in order to guarantee that the agents' states are kept private. We show that under mild conditions each agent's optimization problem converges in mean-square to its unique solution while each agent's state is kept differentially private. A numerical simulation is provided to demonstrate the viability of this approach.

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arXiv2016

Cloud-Enabled Differentially Private Multi-Agent Optimization with Constraints

We present an optimization framework for solving multi-agent nonlinear programs subject to inequality constraints while keeping the agents' state trajectories private. Each agent has an objective function depending only upon its own state and the agents are collectively subject to global constraints. The agents do not directly communicate with each other but instead route messages through a trusted cloud computer. The cloud computer adds noise to data being sent to the agents in accordance with the framework of differential privacy in order to keep each agent's state trajectory private from all other agents and any eavesdroppers. This private problem can be viewed as a stochastic variational inequality and is solved using a projection-based method for solving variational inequalities that resembles a noisy primal-dual gradient algorithm. Convergence of the optimization algorithm in the presence of noise is proven and a quantifiable trade-off between privacy and convergence is extracted from this proof. Simulation results are provided that demonstrate numerical convergence for both $ε$-differential privacy and $(ε, δ)$-differential privacy.

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arXiv2014

Cloud-Based Optimization: A Quasi-Decentralized Approach to Multi-Agent Coordination

New architectures and algorithms are needed to reflect the mixture of local and global information that is available as multi-agent systems connect over the cloud. We present a novel architecture for multi-agent coordination where the cloud is assumed to be able to gather information from all agents, perform centralized computations, and disseminate the results in an intermittent manner. This architecture is used to solve a multi-agent optimization problem in which each agent has a local objective function unknown to the other agents and in which the agents are collectively subject to global inequality constraints. Leveraging the cloud, a dual problem is formulated and solved by finding a saddle point of the associated Lagrangian.

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