arXiv subjects

Ufuk Topcu

Publications and source records attributed to Ufuk Topcu.

At least 235 records · Page 13Linked to original sources

arXiv2021

Convex Optimization for Parameter Synthesis in MDPs

Probabilistic model checking aims to prove whether a Markov decision process (MDP) satisfies a temporal logic specification. The underlying methods rely on an often unrealistic assumption that the MDP is precisely known. Consequently, parametric MDPs (pMDPs) extend MDPs with transition probabilities that are functions over unspecified parameters. The parameter synthesis problem is to compute an instantiation of these unspecified parameters such that the resulting MDP satisfies the temporal logic specification. We formulate the parameter synthesis problem as a quadratically constrained quadratic program (QCQP), which is nonconvex and is NP-hard to solve in general. We develop two approaches that iteratively obtain locally optimal solutions. The first approach exploits the so-called convex-concave procedure (CCP), and the second approach utilizes a sequential convex programming (SCP) method. The techniques improve the runtime and scalability by multiple orders of magnitude compared to black-box CCP and SCP by merging ideas from convex optimization and probabilistic model checking. We demonstrate the approaches on a satellite collision avoidance problem with hundreds of thousands of states and tens of thousands of parameters and their scalability on a wide range of commonly used benchmarks.

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arXiv2021

Probabilistic Control of Heterogeneous Swarms Subject to Graph Temporal Logic Specifications: A Decentralized and Scalable Approach

We develop a probabilistic control algorithm, $\texttt{GTLProCo}$, for swarms of agents with heterogeneous dynamics and objectives, subject to high-level task specifications. The resulting algorithm not only achieves decentralized control of the swarm but also significantly improves scalability over state-of-the-art existing algorithms. Specifically, we study a setting in which the agents move along the nodes of a graph, and the high-level task specifications for the swarm are expressed in a recently-proposed language called graph temporal logic (GTL). By constraining the distribution of the swarm over the nodes of the graph, GTL can specify a wide range of properties, including safety, progress, and response. $\texttt{GTLProCo}$, agnostic to the number of agents comprising the swarm, controls the density distribution of the swarm in a decentralized and probabilistic manner. To this end, it synthesizes a time-varying Markov chain modeling the time evolution of the density distribution under the GTL constraints. We first identify a subset of GTL, namely reach-avoid specifications, for which we can reduce the synthesis of such a Markov chain to either linear or semi-definite programs. Then, in the general case, we formulate the synthesis of the Markov chain as a mixed-integer nonlinear program (MINLP). We exploit the structure of the problem to provide an efficient sequential mixed-integer linear programming scheme with trust regions to solve the MINLP. We empirically demonstrate that our sequential scheme is at least three orders of magnitude faster than off-the-shelf MINLP solvers and illustrate the effectiveness of $\texttt{GTLProCo}$ in several swarm scenarios.

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arXiv2021

Learning Linear Temporal Properties from Noisy Data: A MaxSAT Approach

We address the problem of inferring descriptions of system behavior using Linear Temporal Logic (LTL) from a finite set of positive and negative examples. Most of the existing approaches for solving such a task rely on predefined templates for guiding the structure of the inferred formula. The approaches that can infer arbitrary LTL formulas, on the other hand, are not robust to noise in the data. To alleviate such limitations, we devise two algorithms for inferring concise LTL formulas even in the presence of noise. Our first algorithm infers minimal LTL formulas by reducing the inference problem to a problem in maximum satisfiability and then using off-the-shelf MaxSAT solvers to find a solution. To the best of our knowledge, we are the first to incorporate the usage of MaxSAT solvers for inferring formulas in LTL. Our second learning algorithm relies on the first algorithm to derive a decision tree over LTL formulas based on a decision tree learning algorithm. We have implemented both our algorithms and verified that our algorithms are efficient in extracting concise LTL descriptions even in the presence of noise.

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arXiv2021

Privacy-Utility Trade-Offs Against Limited Adversaries

We study privacy-utility trade-offs where users share privacy-correlated useful information with a service provider to obtain some utility. The service provider is adversarial in the sense that it can infer the users' private information based on the shared useful information. To minimize the privacy leakage while maintaining a desired level of utility, the users carefully perturb the useful information via a probabilistic privacy mapping before sharing it. We focus on the setting in which the adversary attempting an inference attack on the users' privacy has potentially biased information about the statistical correlation between the private and useful variables. This information asymmetry between the users and the limited adversary leads to better privacy guarantees than the case of the omniscient adversary under the same utility requirement. We first identify assumptions on the adversary's information so that the inference costs are well-defined and finite. Then, we characterize the impact of the information asymmetry and show that it increases the inference costs for the adversary. We further formulate the design of the privacy mapping against a limited adversary using a difference of convex functions program and solve it via the concave-convex procedure. When the adversary's information is not precisely available, we adopt a Bayesian view and represent the adversary's information by a probability distribution. In this case, the expected cost for the adversary does not admit a closed-form expression, and we establish and maximize a lower bound of the expected cost. We provide a numerical example regarding a census data set to illustrate the theoretical results.

cs.IT↗

arXiv2021

Robust Training in High Dimensions via Block Coordinate Geometric Median Descent

Geometric median (\textsc{Gm}) is a classical method in statistics for achieving a robust estimation of the uncorrupted data; under gross corruption, it achieves the optimal breakdown point of 0.5. However, its computational complexity makes it infeasible for robustifying stochastic gradient descent (SGD) for high-dimensional optimization problems. In this paper, we show that by applying \textsc{Gm} to only a judiciously chosen block of coordinates at a time and using a memory mechanism, one can retain the breakdown point of 0.5 for smooth non-convex problems, with non-asymptotic convergence rates comparable to the SGD with \textsc{Gm}.

cs.LG↗

arXiv2021

Towards Online Monitoring and Data-driven Control: A Study of Segmentation Algorithms for Laser Powder Bed Fusion Processes

An increasing number of laser powder bed fusion machines use off-axis infrared cameras to improve online monitoring and data-driven control capabilities. However, there is still a severe lack of algorithmic solutions to properly process the infrared images from these cameras that has led to several key limitations: a lack of online monitoring capabilities for the laser tracks, insufficient pre-processing of the infrared images for data-driven methods, and large memory requirements for storing the infrared images. To address these limitations, we study over 30 segmentation algorithms that segment each infrared image into a foreground and background. By evaluating each algorithm based on its segmentation accuracy, computational speed, and spatter detection characteristics, we identify promising algorithmic solutions. The identified algorithms can be readily applied to the laser powder bed fusion machines to address each of the above limitations and thus, significantly improve process control.

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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

Distributed Policy Synthesis of Multi-Agent Systems With Graph Temporal Logic Specifications

We study the distributed synthesis of policies for multi-agent systems to perform \emph{spatial-temporal} tasks. We formalize the synthesis problem as a \emph{factored} Markov decision process subject to \emph{graph temporal logic} specifications. The transition function and task of each agent are functions of the agent itself and its neighboring agents. In this work, we develop another distributed synthesis method, which improves the scalability and runtime by two orders of magnitude compared to our prior work. The synthesis method decomposes the problem into a set of smaller problems, one for each agent by leveraging the structure in the model, and the specifications. We show that the running time of the method is linear in the number of agents. The size of the problem for each agent is exponential only in the number of neighboring agents, which is typically much smaller than the number of agents. We demonstrate the applicability of the method in case studies on disease control, urban security, and search and rescue. The numerical examples show that the method scales to hundreds of agents with hundreds of states per agent and can also handle significantly larger state spaces than our prior work.

cs.MA↗

arXiv2021

Uncertainty-Aware Signal Temporal Logic Inference

Temporal logic inference is the process of extracting formal descriptions of system behaviors from data in the form of temporal logic formulas. The existing temporal logic inference methods mostly neglect uncertainties in the data, which results in limited applicability of such methods in real-world deployments. In this paper, we first investigate the uncertainties associated with trajectories of a system and represent such uncertainties in the form of interval trajectories. We then propose two uncertainty-aware signal temporal logic (STL) inference approaches to classify the undesired behaviors and desired behaviors of a system. Instead of classifying finitely many trajectories, we classify infinitely many trajectories within the interval trajectories. In the first approach, we incorporate robust semantics of STL formulas with respect to an interval trajectory to quantify the margin at which an STL formula is satisfied or violated by the interval trajectory. The second approach relies on the first learning algorithm and exploits the decision tree to infer STL formulas to classify behaviors of a given system. The proposed approaches also work for non-separable data by optimizing the worst-case robustness in inferring an STL formula. Finally, we evaluate the performance of the proposed algorithms in two case studies, where the proposed algorithms show reductions in the computation time by up to four orders of magnitude in comparison with the sampling-based baseline algorithms (for a dataset with 800 sampled trajectories in total).

cs.AI↗

arXiv2021

Entropy Maximization for Partially Observable Markov Decision Processes

We study the problem of synthesizing a controller that maximizes the entropy of a partially observable Markov decision process (POMDP) subject to a constraint on the expected total reward. Such a controller minimizes the predictability of an agent's trajectories to an outside observer while guaranteeing the completion of a task expressed by a reward function. We first prove that an agent with partial observations can achieve an entropy at most as well as an agent with perfect observations. Then, focusing on finite-state controllers (FSCs) with deterministic memory transitions, we show that the maximum entropy of a POMDP is lower bounded by the maximum entropy of the parametric Markov chain (pMC) induced by such FSCs. This relationship allows us to recast the entropy maximization problem as a so-called parameter synthesis problem for the induced pMC. We then present an algorithm to synthesize an FSC that locally maximizes the entropy of a POMDP over FSCs with the same number of memory states. In numerical examples, we illustrate the relationship between the maximum entropy, the number of memory states in the FSC, and the expected reward.

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arXiv2021

Safety-Constrained Learning and Control using Scarce Data and Reciprocal Barriers

We develop a control algorithm that ensures the safety, in terms of confinement in a set, of a system with unknown, 2nd-order nonlinear dynamics. The algorithm establishes novel connections between data-driven and robust, nonlinear control. It is based on data obtained online from the current trajectory and the concept of reciprocal barriers. More specifically, it first uses the obtained data to calculate set-valued functions that over-approximate the unknown dynamic terms. For the second step of the algorithm, we design a robust control scheme that uses these functions as well as reciprocal barriers to render the system forward invariant with respect to the safe set. In addition, we provide an extension of the algorithm that tackles issues of controllability loss incurred by the nullspace of the control-direction matrix. The algorithm removes a series of standard, limiting assumptions considered in the related literature since it does not require global boundedness, growth conditions, or a priori approximations of the unknown dynamics' terms.

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arXiv2021

Efficient Strategy Synthesis for MDPs with Resource Constraints

We consider qualitative strategy synthesis for the formalism called consumption Markov decision processes. This formalism can model dynamics of an agents that operates under resource constraints in a stochastic environment. The presented algorithms work in time polynomial with respect to the representation of the model and they synthesize strategies ensuring that a given set of goal states will be reached (once or infinitely many times) with probability 1 without resource exhaustion. In particular, when the amount of resource becomes too low to safely continue in the mission, the strategy changes course of the agent towards one of a designated set of reload states where the agent replenishes the resource to full capacity; with sufficient amount of resource, the agent attempts to fulfill the mission again. We also present two heuristics that attempt to reduce expected time that the agent needs to fulfill the given mission, a parameter important in practical planning. The presented algorithms were implemented and numerical examples demonstrate (i) the effectiveness (in terms of computation time) of the planning approach based on consumption Markov decision processes and (ii) the positive impact of the two heuristics on planning in a realistic example.

cs.AI↗

arXiv2021

Polynomial-Time Algorithms for Multi-Agent Minimal-Capacity Planning

We study the problem of minimizing the resource capacity of autonomous agents cooperating to achieve a shared task. More specifically, we consider high-level planning for a team of homogeneous agents that operate under resource constraints in stochastic environments and share a common goal: given a set of target locations, ensure that each location will be visited infinitely often by some agent almost surely. We formalize the dynamics of agents by consumption Markov decision processes. In a consumption Markov decision process, the agent has a resource of limited capacity. Each action of the agent may consume some amount of the resource. To avoid exhaustion, the agent can replenish its resource to full capacity in designated reload states. The resource capacity restricts the capabilities of the agent. The objective is to assign target locations to agents, and each agent is only responsible for visiting the assigned subset of target locations repeatedly. Moreover, the assignment must ensure that the agents can carry out their tasks with minimal resource capacity. We reduce the problem of finding target assignments for a team of agents with the lowest possible capacity to an equivalent graph-theoretical problem. We develop an algorithm that solves this graph problem in time that is \emph{polynomial} in the number of agents, target locations, and size of the consumption Markov decision process. We demonstrate the applicability and scalability of the algorithm in a scenario where hundreds of unmanned underwater vehicles monitor hundreds of locations in environments with stochastic ocean currents.

cs.MA↗

arXiv2021

On Controllability and Persistency of Excitation in Data-Driven Control: Extensions of Willems' Fundamental Lemma

Willems' fundamental lemma asserts that all trajectories of a linear time-invariant system can be obtained from a finite number of measured ones, assuming that controllability and a persistency of excitation condition hold. We show that these two conditions can be relaxed. First, we prove that the controllability condition can be replaced by a condition on the controllable subspace, unobservable subspace, and a certain subspace associated with the measured trajectories. Second, we prove that the persistency of excitation requirement can be relaxed if the degree of a certain minimal polynomial is tightly bounded. Our results show that data-driven predictive control using online data is equivalent to model predictive control, even for uncontrollable systems. Moreover, our results significantly reduce the amount of data needed in identifying homogeneous multi-agent systems.

eess.SY↗

arXiv2021

Temporal-Logic-Based Intermittent, Optimal, and Safe Continuous-Time Learning for Trajectory Tracking

In this paper, we develop safe reinforcement-learning-based controllers for systems tasked with accomplishing complex missions that can be expressed as linear temporal logic specifications, similar to those required by search-and-rescue missions. We decompose the original mission into a sequence of tracking sub-problems under safety constraints. We impose the safety conditions by utilizing barrier functions to map the constrained optimal tracking problem in the physical space to an unconstrained one in the transformed space. Furthermore, we develop policies that intermittently update the control signal to solve the tracking sub-problems with reduced burden in the communication and computation resources. Subsequently, an actor-critic algorithm is utilized to solve the underlying Hamilton-Jacobi-Bellman equations. Finally, we support our proposed framework with stability proofs and showcase its efficacy via simulation results.

eess.SY↗

arXiv2021

On-The-Fly Control of Unknown Smooth Systems from Limited Data

We investigate the problem of data-driven, on-the-fly control of systems with unknown nonlinear dynamics where data from only a single finite-horizon trajectory and possibly side information on the dynamics are available. Such side information may include knowledge of the regularity of the dynamics, monotonicity of the states, or decoupling in the dynamics between the states. Specifically, we develop two algorithms, $\texttt{DaTaReach}$ and $\texttt{DaTaControl}$, to over-approximate the reachable set and design control signals for the system on the fly. $\texttt{DaTaReach}$ constructs a differential inclusion that contains the unknown vector field. Then, it computes an over-approximation of the reachable set based on interval Taylor-based methods applied to systems with dynamics described as differential inclusions. $\texttt{DaTaControl}$ enables convex-optimization-based, near-optimal control using the computed over-approximation and the receding-horizon control framework. We provide a bound on its suboptimality and show that more data and side information enable $\texttt{DaTaControl}$ to achieve tighter suboptimality bounds. Finally, we demonstrate the efficacy of $\texttt{DaTaControl}$ over existing approaches on the problems of controlling a unicycle and quadrotor systems.

eess.SY↗

arXiv2021

Collaborative Beamforming Under Localization Errors: A Discrete Optimization Approach

We consider a network of agents that locate themselves in an environment through sensor measurements and aim to transmit a message signal to a base station via collaborative beamforming. The agents' sensor measurements result in localization errors, which degrade the quality of service at the base station due to unknown phase offsets that arise in the agents' communication channels. Assuming that each agent's localization error follows a Gaussian distribution, we study the problem of forming a reliable communication link between the agents and the base station despite the localization errors. In particular, we formulate a discrete optimization problem to choose only a subset of agents to transmit the message signal so that the variance of the signal-to-noise ratio (SNR) received by the base station is minimized while the expected SNR exceeds a desired threshold. When the variances of the localization errors are below a certain threshold characterized in terms of the carrier frequency, we show that greedy algorithms can be used to globally minimize the variance of the received SNR. On the other hand, when some agents have localization errors with large variances, we show that the variance of the received SNR can be locally minimized by exploiting the supermodularity of the mean and variance of the received SNR. In numerical simulations, we demonstrate that the proposed algorithms have the potential to synthesize beamformers orders of magnitude faster than convex optimization-based approaches while achieving comparable performances using less number of agents.

eess.SP↗

arXiv2021

On Minimizing Total Discounted Cost in MDPs Subject to Reachability Constraints

We study the synthesis of a policy in a Markov decision process (MDP) following which an agent reaches a target state in the MDP while minimizing its total discounted cost. The problem combines a reachability criterion with a discounted cost criterion and naturally expresses the completion of a task with probabilistic guarantees and optimal transient performance. We first establish that an optimal policy for the considered formulation may not exist but that there always exists a near-optimal stationary policy. We additionally provide a necessary and sufficient condition for the existence of an optimal policy. We then restrict our attention to stationary deterministic policies and show that the decision problem associated with the synthesis of an optimal stationary deterministic policy is NP-complete. Finally, we provide an exact algorithm based on mixed-integer linear programming and propose an efficient approximation algorithm based on linear programming for the synthesis of an optimal stationary deterministic policy.

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