arXiv subjects

George J. Pappas

Publications and source records attributed to George J. Pappas.

At least 145 records · Page 8Linked to original sources

arXiv2020

Learning Lyapunov Functions for Hybrid Systems

We propose a sampling-based approach to learn Lyapunov functions for a class of discrete-time autonomous hybrid systems that admit a mixed-integer representation. Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical system in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to find a valid Lyapunov function inside a convex set of Lyapunov function candidates. In each iteration, the learner uses a collection of state samples to select a Lyapunov function candidate through a convex program in the parameter space. The verifier then solves a mixed-integer quadratic program in the state space to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates. By designing the learner and the verifier according to the analytic center cutting-plane method from convex optimization, we show that when the set of Lyapunov functions is full-dimensional in the parameter space, our method finds a Lyapunov function in a finite number of steps. We demonstrate our stability analysis method on closed-loop MPC dynamical systems and a ReLU neural network controlled PWA system.

math.OC↗

arXiv2020

Resilient Monotone Sequential Maximization

Applications in machine learning, optimization, and control require the sequential selection of a few system elements, such as sensors, data, or actuators, to optimize the system performance across multiple time steps. However, in failure-prone and adversarial environments, sensors get attacked, data get deleted, and actuators fail. Thence, traditional sequential design paradigms become insufficient and, in contrast, resilient sequential designs that adapt against system-wide attacks, deletions, or failures become important. In general, resilient sequential design problems are computationally hard. Also, even though they often involve objective functions that are monotone and (possibly) submodular, no scalable approximation algorithms are known for their solution. In this paper, we provide the first scalable algorithm, that achieves the following characteristics: system-wide resiliency, i.e., the algorithm is valid for any number of denial-of-service attacks, deletions, or failures; adaptiveness, i.e., at each time step, the algorithm selects system elements based on the history of inflicted attacks, deletions, or failures; and provable approximation performance, i.e., the algorithm guarantees for monotone objective functions a solution close to the optimal. We quantify the algorithm's approximation performance using a notion of curvature for monotone (not necessarily submodular) set functions. Finally, we support our theoretical analyses with simulated experiments, by considering a control-aware sensor scheduling scenario, namely, sensing-constrained robot navigation.

stat.ML↗

arXiv2020

Robust and Adaptive Sequential Submodular Optimization

Emerging applications of control, estimation, and machine learning, ranging from target tracking to decentralized model fitting, pose resource constraints that limit which of the available sensors, actuators, or data can be simultaneously used across time. Therefore, many researchers have proposed solutions within discrete optimization frameworks where the optimization is performed over finite sets. By exploiting notions of discrete convexity, such as submodularity, the researchers have been able to provide scalable algorithms with provable suboptimality bounds. In this paper, we consider such problems but in adversarial environments, where in every step a number of the chosen elements in the optimization is removed due to failures/attacks. Specifically, we consider for the first time a sequential version of the problem that allows us to observe the failures and adapt, while the attacker also adapts to our response. We call the novel problem Robust Sequential submodular Maximization (RSM). Generally, the problem is computationally hard and no scalable algorithm is known for its solution. However, in this paper we propose Robust and Adaptive Maximization (RAM), the first scalable algorithm. RAM runs in an online fashion, adapting in every step to the history of failures. Also, it guarantees a near-optimal performance, even against any number of failures among the used elements. Particularly, RAM has both provable per-instance a priori bounds and tight and/or optimal a posteriori bounds. Finally, we demonstrate RAM's near-optimality in simulations across various application scenarios, along with its robustness against several failure types, from worst-case to random.

math.OC↗

arXiv2020

Reactive Temporal Logic Planning for Multiple Robots in Unknown Occupancy Grid Maps

This paper proposes a new reactive temporal logic planning algorithm for multiple robots that operate in environments with unknown geometry modeled using occupancy grid maps. The robots are equipped with individual sensors that allow them to continuously learn a grid map of the unknown environment using existing Simultaneous Localization and Mapping (SLAM) methods. The goal of the robots is to accomplish complex collaborative tasks, captured by global Linear Temporal Logic (LTL) formulas. The majority of existing LTL planning approaches rely on discrete abstractions of the robot dynamics operating in known environments and, as a result, they cannot be applied to the more realistic scenarios where the environment is initially unknown. In this paper, we address this novel challenge by proposing the first reactive, abstraction-free, and distributed LTL planning algorithm that can be applied for complex mission planning of multiple robots operating in unknown environments. The proposed algorithm is reactive, i.e., planning is adapting to the updated environmental map and abstraction-free as it does not rely on designing abstractions of the robot dynamics. Also, our algorithm is distributed in the sense that the global LTL task is decomposed into single-agent reachability problems constructed online based on the continuously learned map. The proposed algorithm is complete under mild assumptions on the structure of the environment and the sensor models. We provide extensive numerical simulations and hardware experiments that illustrate the theoretical analysis and show that the proposed algorithm can address complex planning tasks for large-scale multi-robot systems in unknown environments.

cs.RO↗

arXiv2020

Learning Lyapunov Functions for Piecewise Affine Systems with Neural Network Controllers

We propose a learning-based method for Lyapunov stability analysis of piecewise affine dynamical systems in feedback with piecewise affine neural network controllers. The proposed method consists of an iterative interaction between a learner and a verifier, where in each iteration, the learner uses a collection of samples of the closed-loop system to propose a Lyapunov function candidate as the solution to a convex program. The learner then queries the verifier, which solves a mixed-integer program to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the stability condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates. We design the learner and the verifier based on the analytic center cutting-plane method, in which the verifier acts as the cutting-plane oracle to refine the set of Lyapunov function candidates. We show that when the set of Lyapunov functions is full-dimensional in the parameter space, the overall procedure finds a Lyapunov function in a finite number of iterations. We demonstrate the utility of the proposed method in searching for quadratic and piecewise quadratic Lyapunov functions.

math.OC↗

arXiv2020

Optimal Algorithms for Submodular Maximization with Distributed Constraints

We consider a class of discrete optimization problems that aim to maximize a submodular objective function subject to a distributed partition matroid constraint. More precisely, we consider a networked scenario in which multiple agents choose actions from local strategy sets with the goal of maximizing a submodular objective function defined over the set of all possible actions. Given this distributed setting, we develop Constraint-Distributed Continuous Greedy (CDCG), a message passing algorithm that converges to the tight $(1-1/e)$ approximation factor of the optimum global solution using only local computation and communication. It is known that a sequential greedy algorithm can only achieve a $1/2$ multiplicative approximation of the optimal solution for this class of problems in the distributed setting. Our framework relies on lifting the discrete problem to a continuous domain and developing a consensus algorithm that achieves the tight $(1-1/e)$ approximation guarantee of the global discrete solution once a proper rounding scheme is applied. We also offer empirical results from a multi-agent area coverage problem to show that the proposed method significantly outperforms the state-of-the-art sequential greedy method.

math.OC↗

arXiv2020

Model-Based Robust Deep Learning: Generalizing to Natural, Out-of-Distribution Data

While deep learning has resulted in major breakthroughs in many application domains, the frameworks commonly used in deep learning remain fragile to artificially-crafted and imperceptible changes in the data. In response to this fragility, adversarial training has emerged as a principled approach for enhancing the robustness of deep learning with respect to norm-bounded perturbations. However, there are other sources of fragility for deep learning that are arguably more common and less thoroughly studied. Indeed, natural variation such as lighting or weather conditions can significantly degrade the accuracy of trained neural networks, proving that such natural variation presents a significant challenge for deep learning. In this paper, we propose a paradigm shift from perturbation-based adversarial robustness toward model-based robust deep learning. Our objective is to provide general training algorithms that can be used to train deep neural networks to be robust against natural variation in data. Critical to our paradigm is first obtaining a model of natural variation which can be used to vary data over a range of natural conditions. Such models may be either known a priori or else learned from data. In the latter case, we show that deep generative models can be used to learn models of natural variation that are consistent with realistic conditions. We then exploit such models in three novel model-based robust training algorithms in order to enhance the robustness of deep learning with respect to the given model. Our extensive experiments show that across a variety of naturally-occurring conditions and across various datasets, deep neural networks trained with our model-based algorithms significantly outperform both standard deep learning algorithms as well as norm-bounded robust deep learning algorithms.

cs.LG↗

arXiv2020

Risk-Constrained Linear-Quadratic Regulators

We propose a new risk-constrained reformulation of the standard Linear Quadratic Regulator (LQR) problem. Our framework is motivated by the fact that the classical (risk-neutral) LQR controller, although optimal in expectation, might be ineffective under relatively infrequent, yet statistically significant (risky) events. To effectively trade between average and extreme event performance, we introduce a new risk constraint, which explicitly restricts the total expected predictive variance of the state penalty by a user-prescribed level. We show that, under rather minimal conditions on the process noise (i.e., finite fourth-order moments), the optimal risk-aware controller can be evaluated explicitly and in closed form. In fact, it is affine relative to the state, and is always internally stable regardless of parameter tuning. Our new risk-aware controller: i) pushes the state away from directions where the noise exhibits heavy tails, by exploiting the third-order moment (skewness) of the noise; ii) inflates the state penalty in riskier directions, where both the noise covariance and the state penalty are simultaneously large. The properties of the proposed risk-aware LQR framework are also illustrated via indicative numerical examples.

eess.SY↗

arXiv2020

Private Weighted Sum Aggregation

As large amounts of data are circulated both from users to a cloud server and between users, there is a critical need for privately aggregating the shared data. This paper considers the problem of private weighted sum aggregation with secret weights, where an aggregator wants to compute the weighted sum of the local data of some agents. Depending on the privacy requirements posed on the weights, there are different secure multi-party computation schemes exploiting the information structure. First, when each agent has a local private value and a local private weight, we review private sum aggregation schemes. Second, we discuss how to extend the previous schemes for when the agents have a local private value, but the aggregator holds the corresponding weights. Third, we treat a more general case where the agents have their local private values, but the weights are known neither by the agents nor by the aggregator; they are generated by a system operator, who wants to keep them private. We give a solution where aggregator obliviousness is achieved, even under collusion between the participants, and we show how to obtain a more efficient communication and computation strategy for multi-dimensional data, by batching the data into fewer ciphertexts. Finally, we implement our schemes and discuss the numerical results and efficiency improvements.

cs.CR↗

arXiv2020

Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes

This paper focuses on the controller synthesis for unknown, nonlinear systems while ensuring safety constraints. Our approach consists of two steps, a learning step that uses Gaussian processes and a controller synthesis step that is based on control barrier functions. In the learning step, we use a data-driven approach utilizing Gaussian processes to learn the unknown control affine nonlinear dynamics together with a statistical bound on the accuracy of the learned model. In the second controller synthesis steps, we develop a systematic approach to compute control barrier functions that explicitly take into consideration the uncertainty of the learned model. The control barrier function not only results in a safe controller by construction but also provides a rigorous lower bound on the probability of satisfaction of the safety specification. Finally, we illustrate the effectiveness of the proposed results by synthesizing a safety controller for a jet engine example.

eess.SY↗

arXiv2020

Control Barrier Functions for Nonholonomic Systems under Risk Signal Temporal Logic Specifications

Temporal logics provide a formalism for expressing complex system specifications. A large body of literature has addressed the verification and the control synthesis problem for deterministic systems under such specifications. For stochastic systems or systems operating in unknown environments, however, only the probability of satisfying a specification has been considered so far, neglecting the risk of not satisfying the specification. Towards addressing this shortcoming, we consider, for the first time, risk metrics, such as (but not limited to) the Conditional Value-at-Risk, and propose risk signal temporal logic. Specifically, we compose risk metrics with stochastic predicates to consider the risk of violating certain spatial specifications. As a particular instance of such stochasticity, we consider control systems in unknown environments and present a determinization of the risk signal temporal logic specification to transform the stochastic control problem into a deterministic one. For unicycle-like dynamics, we then extend our previous work on deterministic time-varying control barrier functions.

math.LO↗

arXiv2020

Zeroth-order Deterministic Policy Gradient

Deterministic Policy Gradient (DPG) removes a level of randomness from standard randomized-action Policy Gradient (PG), and demonstrates substantial empirical success for tackling complex dynamic problems involving Markov decision processes. At the same time, though, DPG loses its ability to learn in a model-free (i.e., actor-only) fashion, frequently necessitating the use of critics in order to obtain consistent estimates of the associated policy-reward gradient. In this work, we introduce Zeroth-order Deterministic Policy Gradient (ZDPG), which approximates policy-reward gradients via two-point stochastic evaluations of the $Q$-function, constructed by properly designed low-dimensional action-space perturbations. Exploiting the idea of random horizon rollouts for obtaining unbiased estimates of the $Q$-function, ZDPG lifts the dependence on critics and restores true model-free policy learning, while enjoying built-in and provable algorithmic stability. Additionally, we present new finite sample complexity bounds for ZDPG, which improve upon existing results by up to two orders of magnitude. Our findings are supported by several numerical experiments, which showcase the effectiveness of ZDPG in a practical setting, and its advantages over both PG and Baseline PG.

cs.LG↗

arXiv2020

Learning to Track Dynamic Targets in Partially Known Environments

We solve active target tracking, one of the essential tasks in autonomous systems, using a deep reinforcement learning (RL) approach. In this problem, an autonomous agent is tasked with acquiring information about targets of interests using its onboard sensors. The classical challenges in this problem are system model dependence and the difficulty of computing information-theoretic cost functions for a long planning horizon. RL provides solutions for these challenges as the length of its effective planning horizon does not affect the computational complexity, and it drops the strong dependency of an algorithm on system models. In particular, we introduce Active Tracking Target Network (ATTN), a unified RL policy that is capable of solving major sub-tasks of active target tracking -- in-sight tracking, navigation, and exploration. The policy shows robust behavior for tracking agile and anomalous targets with a partially known target model. Additionally, the same policy is able to navigate in obstacle environments to reach distant targets as well as explore the environment when targets are positioned in unexpected locations.

cs.LG↗

arXiv2020

LQG Control and Sensing Co-Design

We investigate a Linear-Quadratic-Gaussian (LQG) control and sensing co-design problem, where one jointly designs sensing and control policies. We focus on the realistic case where the sensing design is selected among a finite set of available sensors, where each sensor is associated with a different cost (e.g., power consumption). We consider two dual problem instances: sensing-constrained LQG control, where one maximizes control performance subject to a sensor cost budget, and minimum-sensing LQG control, where one minimizes sensor cost subject to performance constraints. We prove no polynomial time algorithm guarantees across all problem instances a constant approximation factor from the optimal. Nonetheless, we present the first polynomial time algorithms with per-instance suboptimality guarantees. To this end, we leverage a separation principle, that partially decouples the design of sensing and control. Then, we frame LQG co-design as the optimization of approximately supermodular set functions; we develop novel algorithms to solve the problems; and we prove original results on the performance of the algorithms, and establish connections between their suboptimality and control-theoretic quantities. We conclude the paper by discussing two applications, namely, sensing-constrained formation control and resource-constrained robot navigation.

math.OC↗

arXiv2020

Sensor Placement for Optimal Kalman Filtering: Fundamental Limits, Submodularity, and Algorithms

In this paper, we focus on sensor placement in linear dynamic estimation, where the objective is to place a small number of sensors in a system of interdependent states so to design an estimator with a desired estimation performance. In particular, we consider a linear time-variant system that is corrupted with process and measurement noise, and study how the selection of its sensors affects the estimation error of the corresponding Kalman filter over a finite observation interval. Our contributions are threefold: First, we prove that the minimum mean square error of the Kalman filter decreases only linearly as the number of sensors increases. That is, adding extra sensors so to reduce this estimation error is ineffective, a fundamental design limit. Similarly, we prove that the number of sensors grows linearly with the system's size for fixed minimum mean square error and number of output measurements over an observation interval; this is another fundamental limit, especially for systems where the system's size is large. Second, we prove that the logdet of the error covariance of the Kalman filter, which captures the volume of the corresponding confidence ellipsoid, with respect to the system's initial condition and process noise is a supermodular and non-increasing set function in the choice of the sensor set. Therefore, it exhibits the diminishing returns property. Third, we provide efficient approximation algorithms that select a small number sensors so to optimize the Kalman filter with respect to this estimation error ---the worst-case performance guarantees of these algorithms are provided as well. Finally, we illustrate the efficiency of our algorithms using the problem of surface-based monitoring of CO2 sequestration sites studied in Weimer et al. (2008).

math.OC↗

arXiv2020

Sensing-Constrained LQG Control

Linear-Quadratic-Gaussian (LQG) control is concerned with the design of an optimal controller and estimator for linear Gaussian systems with imperfect state information. Standard LQG assumes the set of sensor measurements, to be fed to the estimator, to be given. However, in many problems, arising in networked systems and robotics, one may not be able to use all the available sensors, due to power or payload constraints, or may be interested in using the smallest subset of sensors that guarantees the attainment of a desired control goal. In this paper, we introduce the sensing-constrained LQG control problem, in which one has to jointly design sensing, estimation, and control, under given constraints on the resources spent for sensing. We focus on the realistic case in which the sensing strategy has to be selected among a finite set of possible sensing modalities. While the computation of the optimal sensing strategy is intractable, we present the first scalable algorithm that computes a near-optimal sensing strategy with provable sub-optimality guarantees. To this end, we show that a separation principle holds, which allows the design of sensing, estimation, and control policies in isolation. We conclude the paper by discussing two applications of sensing-constrained LQG control, namely, sensing-constrained formation control and resource-constrained robot navigation.

math.OC↗

arXiv2020

Sample Complexity of Kalman Filtering for Unknown Systems

In this paper, we consider the task of designing a Kalman Filter (KF) for an unknown and partially observed autonomous linear time invariant system driven by process and sensor noise. To do so, we propose studying the following two step process: first, using system identification tools rooted in subspace methods, we obtain coarse finite-data estimates of the state-space parameters and Kalman gain describing the autonomous system; and second, we use these approximate parameters to design a filter which produces estimates of the system state. We show that when the system identification step produces sufficiently accurate estimates, or when the underlying true KF is sufficiently robust, that a Certainty Equivalent (CE) KF, i.e., one designed using the estimated parameters directly, enjoys provable sub-optimality guarantees. We further show that when these conditions fail, and in particular, when the CE KF is marginally stable (i.e., has eigenvalues very close to the unit circle), that imposing additional robustness constraints on the filter leads to similar sub-optimality guarantees. We further show that with high probability, both the CE and robust filters have mean prediction error bounded by $\tilde O(1/\sqrt{N})$, where $N$ is the number of data points collected in the system identification step. To the best of our knowledge, these are the first end-to-end sample complexity bounds for the Kalman Filtering of an unknown system.

eess.SY↗

arXiv2020

Technical Report: Reactive Semantic Planning in Unexplored Semantic Environments Using Deep Perceptual Feedback

This paper presents a reactive planning system that enriches the topological representation of an environment with a tightly integrated semantic representation, achieved by incorporating and exploiting advances in deep perceptual learning and probabilistic semantic reasoning. Our architecture combines object detection with semantic SLAM, affording robust, reactive logical as well as geometric planning in unexplored environments. Moreover, by incorporating a human mesh estimation algorithm, our system is capable of reacting and responding in real time to semantically labeled human motions and gestures. New formal results allow tracking of suitably non-adversarial moving targets, while maintaining the same collision avoidance guarantees. We suggest the empirical utility of the proposed control architecture with a numerical study including comparisons with a state-of-the-art dynamic replanning algorithm, and physical implementation on both a wheeled and legged platform in different settings with both geometric and semantic goals.

cs.RO↗