arXiv subjects

Matthew Hale

Publications and source records attributed to Matthew Hale.

At least 55 records · Page 3Linked to original sources

arXiv2023

Characterizing Compositionality of LQR from the Categorical Perspective

Composing systems is a fundamental concept in modern control systems, yet it remains challenging to formally analyze how controllers designed for individual subsystems can differ from controllers designed for the composition of those subsystems. To address this challenge, we propose a novel approach to composing control systems based on resource sharing machines, a concept from applied category theory. We use resource sharing machines to investigate the differences between (i) the linear-quadratic regulator (LQR) designed directly for a composite system and (ii) the LQR that is attained through the composition of LQRs designed for each subsystem. We first establish novel formalisms to compose LQR control designs using resource sharing machines. Then we develop new sufficient conditions to guarantee that the LQR designed for a composite system is equal to the LQR attained through composition of LQRs for its subsystems. In addition, we reduce the developed condition to that of checking the controllability and observability of a certain linear, time-invariant system, which provides a simple, computationally efficient procedure for evaluating the equivalence of controllers for composed systems.

eess.SY↗

arXiv2023

DOMINO: Domain-aware Loss for Deep Learning Calibration

Deep learning has achieved the state-of-the-art performance across medical imaging tasks; however, model calibration is often not considered. Uncalibrated models are potentially dangerous in high-risk applications since the user does not know when they will fail. Therefore, this paper proposes a novel domain-aware loss function to calibrate deep learning models. The proposed loss function applies a class-wise penalty based on the similarity between classes within a given target domain. Thus, the approach improves the calibration while also ensuring that the model makes less risky errors even when incorrect. The code for this software is available at https://github.com/lab-smile/DOMINO.

cs.CV↗

arXiv2023

Differential Privacy in Cooperative Multiagent Planning

Privacy-aware multiagent systems must protect agents' sensitive data while simultaneously ensuring that agents accomplish their shared objectives. Towards this goal, we propose a framework to privatize inter-agent communications in cooperative multiagent decision-making problems. We study sequential decision-making problems formulated as cooperative Markov games with reach-avoid objectives. We apply a differential privacy mechanism to privatize agents' communicated symbolic state trajectories, and then we analyze tradeoffs between the strength of privacy and the team's performance. For a given level of privacy, this tradeoff is shown to depend critically upon the total correlation among agents' state-action processes. We synthesize policies that are robust to privacy by reducing the value of the total correlation. Numerical experiments demonstrate that the team's performance under these policies decreases by only 3 percent when comparing private versus non-private implementations of communication. By contrast, the team's performance decreases by roughly 86 percent when using baseline policies that ignore total correlation and only optimize team performance.

cs.MA↗

arXiv2023

Distributed Reproduction Numbers of Networked Epidemics

Reproduction numbers are widely used for the estimation and prediction of epidemic spreading processes over networks. However, reproduction numbers do not enable estimation and prediction in individual communities within networks, and they can be difficult to compute due to the aggregation of infection data that is required to do so. Therefore, in this work we propose a novel concept of distributed reproduction numbers to capture the spreading behaviors of each entity in the network, and we show how to compute them using certain parameters in networked SIS and SIR epidemic models. We use distributed reproduction numbers to derive new conditions under which an outbreak can occur. These conditions are then used to derive new conditions for the existence, uniqueness, and stability of equilibrium states. Finally, in simulation we use synthetic infection data to illustrate how distributed reproduction numbers provide more fine-grained analyses of networked spreading processes than ordinary reproduction numbers.

eess.SY↗

arXiv2022

Fast Verification of Control Barrier Functions via Linear Programming

Control barrier functions are a popular method of ensuring system safety, and these functions can be used to enforce invariance of a set under the dynamics of a system. A control barrier function must have certain properties, and one must both formulate a candidate control barrier function and verify that it does indeed satisfy the required properties. Targeting the latter problem, this paper presents a method of verifying any finite number of candidate control barrier functions with linear programming. We first apply techniques from real algebraic geometry to formulate verification problem statements that are solvable numerically. Typically, semidefinite programming is used to verify candidate control barrier functions, but this does not always scale well. Therefore, we next apply a method of inner-approximating the set of sums of squares polynomials that significantly reduces the computational complexity of these verification problems by transcribing them to linear programs. We give explicit forms for the resulting linear programs, and simulation results for a satellite inspection problem show that the computation time needed for verification can be reduced by more than 95%.

eess.SY↗

arXiv2022

The Bounded Gaussian Mechanism for Differential Privacy

The Gaussian mechanism is one differential privacy mechanism commonly used to protect numerical data. However, it may be ill-suited to some applications because it has unbounded support and thus can produce invalid numerical answers to queries, such as negative ages or human heights in the tens of meters. One can project such private values onto valid ranges of data, though such projections lead to the accumulation of private query responses at the boundaries of such ranges, thereby harming accuracy. Motivated by the need for both privacy and accuracy over bounded domains, we present a bounded Gaussian mechanism for differential privacy, which has support only on a given region. We present both univariate and multivariate versions of this mechanism and illustrate a significant reduction in variance relative to comparable existing work.

cs.CR↗

arXiv2022

Node and Edge Differential Privacy for Graph Laplacian Spectra: Mechanisms and Scaling Laws

This paper develops a framework for privatizing the spectrum of the graph Laplacian of an undirected graph using differential privacy. We consider two privacy formulations. The first obfuscates the presence of edges in the graph and the second obfuscates the presence of nodes. We compare these two privacy formulations and show that the privacy formulation that considers edges is better suited to most engineering applications. We use the bounded Laplace mechanism to provide differential privacy to the eigenvalues of a graph Laplacian, and we pay special attention to the algebraic connectivity, which is the Laplacian's second smallest eigenvalue. Analytical bounds are presented on the the accuracy of the mechanisms and on certain graph properties computed with private spectra. A suite of numerical examples confirms the accuracy of private spectra in practice.

math.OC↗

arXiv2022

Autonomous Satellite Rendezvous and Proximity Operations with Time-Constrained Sub-Optimal Model Predictive Control

This paper presents a time-constrained model predictive control strategy for the 6 degree-of-freedom (6DOF) autonomous rendezvous and docking problem between a controllable "deputy" spacecraft and an uncontrollable "chief" spacecraft. The control strategy accounts for computational time constraints due to limited onboard processing speed. The translational dynamics model is derived from the Clohessy-Wiltshire equations and the angular dynamics are modeled on gas jet actuation about the deputy's center of mass. Simulation results are shown to achieve the docking configuration under computational time constraints by limiting the number of allowed algorithm iterations when computing each input. Specifically, we show that upwards of 90% of computations can be eliminated from a model predictive control implementation without significantly harming control performance.

math.OC↗

arXiv2022

Technical Report: Distributed Asynchronous Large-Scale Mixed-Integer Linear Programming via Saddle Point Computation

We solve large-scale mixed-integer linear programs (MILPs) via distributed asynchronous saddle point computation. This is motivated by the MILPs being able to model problems in multi-agent autonomy, e.g., task assignment problems and trajectory planning with collision avoidance constraints in multi-robot systems. To solve a MILP, we relax it with a nonlinear program approximation whose accuracy tightens as the number of agents increases relative to the number of coupled constraints. Next, we form an equivalent Lagrangian saddle point problem, and then regularize the Lagrangian in both the primal and dual spaces to create a regularized Lagrangian that is strongly-convex-strongly-concave. We then develop a parallelized algorithm to compute saddle points of the regularized Lagrangian. This algorithm partitions problems into blocks, which are either scalars or sub-vectors of the primal or dual decision variables, and it is shown to tolerate asynchrony in the computations and communications of primal and dual variables. Suboptimality bounds and convergence rates are presented for convergence to a saddle point. The suboptimality bound includes (i) the regularization error induced by regularizing the Lagrangian and (ii) the suboptimality gap between solutions to the original MILP and its relaxed form. Simulation results illustrate these theoretical developments in practice, and show that relaxation and regularization together have only a mild impact on the quality of solution obtained.

math.OC↗

arXiv2022

DOMINO: Domain-aware Model Calibration in Medical Image Segmentation

Model calibration measures the agreement between the predicted probability estimates and the true correctness likelihood. Proper model calibration is vital for high-risk applications. Unfortunately, modern deep neural networks are poorly calibrated, compromising trustworthiness and reliability. Medical image segmentation particularly suffers from this due to the natural uncertainty of tissue boundaries. This is exasperated by their loss functions, which favor overconfidence in the majority classes. We address these challenges with DOMINO, a domain-aware model calibration method that leverages the semantic confusability and hierarchical similarity between class labels. Our experiments demonstrate that our DOMINO-calibrated deep neural networks outperform non-calibrated models and state-of-the-art morphometric methods in head image segmentation. Our results show that our method can consistently achieve better calibration, higher accuracy, and faster inference times than these methods, especially on rarer classes. This performance is attributed to our domain-aware regularization to inform semantic model calibration. These findings show the importance of semantic ties between class labels in building confidence in deep learning models. The framework has the potential to improve the trustworthiness and reliability of generic medical image segmentation models. The code for this article is available at: https://github.com/lab-smile/DOMINO.

eess.IV↗

arXiv2022

Differential Privacy for Symbolic Systems with Application to Markov Chains

Data-driven systems are gathering increasing amounts of data from users, and sensitive user data requires privacy protections. In some cases, the data gathered is non-numerical or symbolic, and conventional approaches to privacy, e.g., adding noise, do not apply, though such systems still require privacy protections. Accordingly, we present a novel differential privacy framework for protecting trajectories generated by symbolic systems. These trajectories can be represented as words or strings over a finite alphabet. We develop new differential privacy mechanisms that approximate a sensitive word using a random word that is likely to be near it. An offline mechanism is implemented efficiently using a Modified Hamming Distance Automaton to generate whole privatized output words over a finite time horizon. Then, an online mechanism is implemented by taking in a sensitive symbol and generating a randomized output symbol at each timestep. This work is extended to Markov chains to generate differentially private state sequences that a given Markov chain could have produced. Statistical accuracy bounds are developed to quantify the accuracy of these mechanisms, and numerical results validate the accuracy of these techniques for strings of English words.

cs.CR↗

arXiv2022

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 construct a family of explicit counterexamples to show the need to eliminate asynchronous communication of dual variables, though the other forms of asynchrony are permitted, all without requiring bounds on delays. A first-order primal-dual 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 include an "asynchrony penalty" that we quantify and present ways to mitigate. Numerical results illustrate these developments.

math.OC↗

arXiv2022

Differentially Private LQ Control

As multi-agent systems proliferate and share more user data, new approaches are needed to protect sensitive data while still enabling system operation. To address this need, this paper presents a private multi-agent LQ control framework. Agents' state trajectories can be sensitive and we therefore protect them using differential privacy. We quantify the impact of privacy along three dimensions: the amount of information shared under privacy, the control-theoretic cost of privacy, and the tradeoffs between privacy and performance. These analyses are done in conventional control-theoretic terms, which we use to develop guidelines for calibrating privacy as a function of system parameters. Numerical results indicate that system performance remains within desirable ranges, even under strict privacy requirements.

math.OC↗

arXiv2021

Technical Report: A Totally Asynchronous Algorithm for Tracking Solutions to Time-Varying Convex Optimization Problems

This paper presents a decentralized algorithm for a team of agents to track time-varying fixed points that are the solutions to time-varying convex optimization problems. The algorithm is first-order, and it allows for total asynchrony in the communications and computations of all agents, i.e., all such operations can occur with arbitrary timing and arbitrary (finite) delays. Convergence rates are computed in terms of the communications and computations that agents execute, without specifying when they must occur. These rates apply to convergence to the minimum of each individual objective function, as well as agents' long-run behavior as their objective functions change. Then, to improve the usage of limited communication and computation resources, we optimize the timing of agents' operations relative to changes in their objective functions to minimize total fixed point tracking error over time. Simulation results are presented to illustrate these developments in practice and empirically assess their robustness to uncertainties in agents' update laws.

math.OC↗

arXiv2021

Exponentially Converging Distributed Gradient Descent with Intermittent Communication via Hybrid Methods

We present a hybrid systems framework for multi-agent optimization in which agents execute computations in continuous time and communicate in discrete time. The optimization algorithm is a hybrid version of parallelized coordinate descent. Agents implement a sample-and-hold strategy in which gradients are computed at communication times and held constant during flows between communications. Completeness of maximal solutions under these hybrid dynamics is established. Under assumptions of smoothness and strong convexity, we show that this system exponentially converges to the minimizer of an objective function. Simulation results illustrate this convergence rate.

math.OC↗

arXiv2021

Privacy-Preserving Kickstarting Deep Reinforcement Learning with Privacy-Aware Learners

Kickstarting deep reinforcement learning algorithms facilitate a teacher-student relationship among the agents and allow for a well-performing teacher to share demonstrations with a student to expedite the student's training. However, despite the known benefits, the demonstrations may contain sensitive information about the teacher's training data and existing kickstarting methods do not take any measures to protect it. Therefore, we use the framework of differential privacy to develop a mechanism that securely shares the teacher's demonstrations with the student. The mechanism allows for the teacher to decide upon the accuracy of its demonstrations with respect to the privacy budget that it consumes, thereby granting the teacher full control over its data privacy. We then develop a kickstarted deep reinforcement learning algorithm for the student that is privacy-aware because we calibrate its objective with the parameters of the teacher's privacy mechanism. The privacy-aware design of the algorithm makes it possible to kickstart the student's learning despite the perturbations induced by the privacy mechanism. From numerical experiments, we highlight three empirical results: (i) the algorithm succeeds in expediting the student's learning, (ii) the student converges to a performance level that was not possible without the demonstrations, and (iii) the student maintains its enhanced performance even after the teacher stops sharing useful demonstrations due to its privacy budget constraints.

cs.LG↗

arXiv2021

Asynchronous Parallel Nonconvex Optimization Under the Polyak-Lojasiewicz Condition

Communication delays and synchronization are major bottlenecks for parallel computing, and tolerating asynchrony is therefore crucial for accelerating parallel computation. Motivated by optimization problems that do not satisfy convexity assumptions, we present an asynchronous block coordinate descent algorithm for nonconvex optimization problems whose objective functions satisfy the Polyak-Lojasiewicz condition. This condition is a generalization of strong convexity to nonconvex problems and requires neither convexity nor uniqueness of minimizers. Under only assumptions of mild smoothness of objective functions and bounded delays, we prove that a linear convergence rate is obtained. Numerical experiments for logistic regression problems are presented to illustrate the impact of asynchrony upon convergence.

math.OC↗

arXiv2021

Edge Differential Privacy for Algebraic Connectivity of Graphs

Graphs are the dominant formalism for modeling multi-agent systems. The algebraic connectivity of a graph is particularly important because it provides the convergence rates of consensus algorithms that underlie many multi-agent control and optimization techniques. However, sharing the value of algebraic connectivity can inadvertently reveal sensitive information about the topology of a graph, such as connections in social networks. Therefore, in this work we present a method to release a graph's algebraic connectivity under a graph-theoretic form of differential privacy, called edge differential privacy. Edge differential privacy obfuscates differences among graphs' edge sets and thus conceals the absence or presence of sensitive connections therein. We provide privacy with bounded Laplace noise, which improves accuracy relative to conventional unbounded noise. The private algebraic connectivity values are analytically shown to provide accurate estimates of consensus convergence rates, as well as accurate bounds on the diameter of a graph and the mean distance between its nodes. Simulation results confirm the utility of private algebraic connectivity in these contexts.

cs.CR↗