arXiv subjects

S. Shankar Sastry

Publications and source records attributed to S. Shankar Sastry.

At least 55 records · Page 3Linked to original sources

arXiv2018

A New Solution Concept and Family of Relaxations for Hybrid Dynamical Systems

We introduce a holistic framework for the analysis, approximation and control of the trajectories of hybrid dynamical systems which display event-triggered discrete jumps in the continuous state. We begin by demonstrating how to explicitly represent the dynamics of this class of systems using a single piecewise-smooth vector field defined on a manifold, and then employ Filippov's solution concept to describe the trajectories of the system. The resulting \emph{hybrid Filippov solutions} greatly simplify the mathematical description of hybrid executions, providing a unifying solution concept with which to work. Extending previous efforts to regularize piecewise-smooth vector fields, we then introduce a parameterized family of smooth control systems whose trajectories are used to approximate the hybrid Filippov solution numerically. The two solution concepts are shown to agree in the limit, under mild regularity conditions.

math.DS↗

arXiv2018

Hierarchical Game-Theoretic Planning for Autonomous Vehicles

The actions of an autonomous vehicle on the road affect and are affected by those of other drivers, whether overtaking, negotiating a merge, or avoiding an accident. This mutual dependence, best captured by dynamic game theory, creates a strong coupling between the vehicle's planning and its predictions of other drivers' behavior, and constitutes an open problem with direct implications on the safety and viability of autonomous driving technology. Unfortunately, dynamic games are too computationally demanding to meet the real-time constraints of autonomous driving in its continuous state and action space. In this paper, we introduce a novel game-theoretic trajectory planning algorithm for autonomous driving, that enables real-time performance by hierarchically decomposing the underlying dynamic game into a long-horizon "strategic" game with simplified dynamics and full information structure, and a short-horizon "tactical" game with full dynamics and a simplified information structure. The value of the strategic game is used to guide the tactical planning, implicitly extending the planning horizon, pushing the local trajectory optimization closer to global solutions, and, most importantly, quantitatively accounting for the autonomous vehicle and the human driver's ability and incentives to influence each other. In addition, our approach admits non-deterministic models of human decision-making, rather than relying on perfectly rational predictions. Our results showcase richer, safer, and more effective autonomous behavior in comparison to existing techniques.

cs.RO↗

arXiv2018

Step Size Matters in Deep Learning

Training a neural network with the gradient descent algorithm gives rise to a discrete-time nonlinear dynamical system. Consequently, behaviors that are typically observed in these systems emerge during training, such as convergence to an orbit but not to a fixed point or dependence of convergence on the initialization. Step size of the algorithm plays a critical role in these behaviors: it determines the subset of the local optima that the algorithm can converge to, and it specifies the magnitude of the oscillations if the algorithm converges to an orbit. To elucidate the effects of the step size on training of neural networks, we study the gradient descent algorithm as a discrete-time dynamical system, and by analyzing the Lyapunov stability of different solutions, we show the relationship between the step size of the algorithm and the solutions that can be obtained with this algorithm. The results provide an explanation for several phenomena observed in practice, including the deterioration in the training error with increased depth, the hardness of estimating linear mappings with large singular values, and the distinct performance of deep residual networks.

cs.LG↗

arXiv2018

Generating Plans that Predict Themselves

Collaboration requires coordination, and we coordinate by anticipating our teammates' future actions and adapting to their plan. In some cases, our teammates' actions early on can give us a clear idea of what the remainder of their plan is, i.e. what action sequence we should expect. In others, they might leave us less confident, or even lead us to the wrong conclusion. Our goal is for robot actions to fall in the first category: we want to enable robots to select their actions in such a way that human collaborators can easily use them to correctly anticipate what will follow. While previous work has focused on finding initial plans that convey a set goal, here we focus on finding two portions of a plan such that the initial portion conveys the final one. We introduce $t$-\ACty{}: a measure that quantifies the accuracy and confidence with which human observers can predict the remaining robot plan from the overall task goal and the observed initial $t$ actions in the plan. We contribute a method for generating $t$-predictable plans: we search for a full plan that accomplishes the task, but in which the first $t$ actions make it as easy as possible to infer the remaining ones. The result is often different from the most efficient plan, in which the initial actions might leave a lot of ambiguity as to how the task will be completed. Through an online experiment and an in-person user study with physical robots, we find that our approach outperforms a traditional efficiency-based planner in objective and subjective collaboration metrics.

cs.RO↗

arXiv2018

Goal Inference Improves Objective and Perceived Performance in Human-Robot Collaboration

The study of human-robot interaction is fundamental to the design and use of robotics in real-world applications. Robots will need to predict and adapt to the actions of human collaborators in order to achieve good performance and improve safety and end-user adoption. This paper evaluates a human-robot collaboration scheme that combines the task allocation and motion levels of reasoning: the robotic agent uses Bayesian inference to predict the next goal of its human partner from his or her ongoing motion, and re-plans its own actions in real time. This anticipative adaptation is desirable in many practical scenarios, where humans are unable or unwilling to take on the cognitive overhead required to explicitly communicate their intent to the robot. A behavioral experiment indicates that the combination of goal inference and dynamic task planning significantly improves both objective and perceived performance of the human-robot team. Participants were highly sensitive to the differences between robot behaviors, preferring to work with a robot that adapted to their actions over one that did not.

cs.RO↗

arXiv2018

Pragmatic-Pedagogic Value Alignment

As intelligent systems gain autonomy and capability, it becomes vital to ensure that their objectives match those of their human users; this is known as the value-alignment problem. In robotics, value alignment is key to the design of collaborative robots that can integrate into human workflows, successfully inferring and adapting to their users' objectives as they go. We argue that a meaningful solution to value alignment must combine multi-agent decision theory with rich mathematical models of human cognition, enabling robots to tap into people's natural collaborative capabilities. We present a solution to the cooperative inverse reinforcement learning (CIRL) dynamic game based on well-established cognitive models of decision making and theory of mind. The solution captures a key reciprocity relation: the human will not plan her actions in isolation, but rather reason pedagogically about how the robot might learn from them; the robot, in turn, can anticipate this and interpret the human's actions pragmatically. To our knowledge, this work constitutes the first formal analysis of value alignment grounded in empirically validated cognitive models.

cs.AI↗

arXiv2017

On the Relaxation of Hybrid Dynamical Systems

Hybrid dynamical systems have proven to be a powerful modeling abstraction, yet fundamental questions regarding the dynamical properties of these systems remain. In this paper, we develop a novel class of relaxations which we use to recover a number of classic systems theoretic properties for hybrid systems, such as existence and uniqueness of trajectories, even past the point of Zeno. Our relaxations also naturally give rise to a class of provably convergent numerical approximations, capable of simulating through Zeno. Using our methods, we are also able to perform sensitivity analysis about nominal trajectories undergoing a discrete transition -- a technique with many practical applications, such as assessing the stability of periodic orbits.

math.DS↗

arXiv2017

A Multi-Armed Bandit Approach for Online Expert Selection in Markov Decision Processes

We formulate a multi-armed bandit (MAB) approach to choosing expert policies online in Markov decision processes (MDPs). Given a set of expert policies trained on a state and action space, the goal is to maximize the cumulative reward of our agent. The hope is to quickly find the best expert in our set. The MAB formulation allows us to quantify the performance of an algorithm in terms of the regret incurred from not choosing the best expert from the beginning. We first develop the theoretical framework for MABs in MDPs, and then present a basic regret decomposition identity. We then adapt the classical Upper Confidence Bounds algorithm to the problem of choosing experts in MDPs and prove that the expected regret grows at worst at a logarithmic rate. Lastly, we validate the theory on a small MDP.

eess.SY↗

arXiv2017

Robust, Informative Human-in-the-Loop Predictions via Empirical Reachable Sets

In order to develop provably safe human-in-the-loop systems, accurate and precise models of human behavior must be developed. In the case of intelligent vehicles, one can imagine the need for predicting driver behavior to develop minimally invasive active safety systems or to safely interact with other vehicles on the road. We present a optimization based method for approximating the stochastic reachable set for human-in-the-loop systems. This method identifies the most precise subset of states that a human driven vehicle may enter, given some dataset of observed trajectories. We phrase this problem as a mixed integer linear program, which can be solved using branch and bound methods. The resulting model uncovers the most representative subset that encapsulates the likely trajectories, up to some probability threshold, by optimally rejecting outliers in the dataset. This tool provides set predictions consisting of trajectories observed from the nonlinear dynamics and behaviors of the human driven car, and can account for modes of behavior, like the driver state or intent. This allows us to predict driving behavior over long time horizons with high accuracy. By using this realistic data and flexible algorithm, a precise and accurate driver model can be developed to capture likely behaviors. The resulting prediction can be tailored to an individual for use in semi-autonomous frameworks or generally applied for autonomous planning in interactive maneuvers.

eess.SY↗

arXiv2017

A Robust Utility Learning Framework via Inverse Optimization

In many smart infrastructure applications flexibility in achieving sustainability goals can be gained by engaging end-users. However, these users often have heterogeneous preferences that are unknown to the decision-maker tasked with improving operational efficiency. Modeling user interaction as a continuous game between non-cooperative players, we propose a robust parametric utility learning framework that employs constrained feasible generalized least squares estimation with heteroskedastic inference. To improve forecasting performance, we extend the robust utility learning scheme by employing bootstrapping with bagging, bumping, and gradient boosting ensemble methods. Moreover, we estimate the noise covariance which provides approximated correlations between players which we leverage to develop a novel correlated utility learning framework. We apply the proposed methods both to a toy example arising from Bertrand-Nash competition between two firms as well as to data from a social game experiment designed to encourage energy efficient behavior amongst smart building occupants. Using occupant voting data for shared resources such as lighting, we simulate the game defined by the estimated utility functions to demonstrate the performance of the proposed methods.

cs.GT↗

arXiv2017

Statistical Estimation with Strategic Data Sources in Competitive Settings

In this paper, we introduce a preliminary model for interactions in the data market. Recent research has shown ways in which a data aggregator can design mechanisms for users to ensure the quality of data, even in situations where the users are effort-averse (i.e. prefer to submit lower-quality estimates) and the data aggregator cannot observe the effort exerted by the users (i.e. the contract suffers from the principal-agent problem). However, we have shown that these mechanisms often break down in more realistic models, where multiple data aggregators are in competition. Under minor assumptions on the properties of the statistical estimators in use by data aggregators, we show that there is either no Nash equilibrium, or there is an infinite number of Nash equilibrium. In the latter case, there is a fundamental ambiguity in who bears the burden of incentivizing different data sources. We are also able to calculate the price of anarchy, which measures how much social welfare is lost between the Nash equilibrium and the social optimum, i.e. between non-cooperative strategic play and cooperation.

cs.GT↗

arXiv2017

Optimal Causal Imputation for Control

The widespread applicability of analytics in cyber-physical systems has motivated research into causal inference methods. Predictive estimators are not sufficient when analytics are used for decision making; rather, the flow of causal effects must be determined. Generally speaking, these methods focus on estimation of a causal structure from experimental data. In this paper, we consider the dual problem: we fix the causal structure and optimize over causal imputations to achieve desirable system behaviors for a minimal imputation cost. First, we present the optimal causal imputation problem, and then we analyze the problem in two special cases: 1) when the causal imputations can only impute to a fixed value, 2) when the causal structure has linear dynamics with additive Gaussian noise. This optimal causal imputation framework serves to bridge the gap between causal structures and control.

eess.SY↗

arXiv2016

Privacy-Enhanced Architecture for Occupancy-based HVAC Control

Large-scale sensing and actuation infrastructures have allowed buildings to achieve significant energy savings; at the same time, these technologies introduce significant privacy risks that must be addressed. In this paper, we present a framework for modeling the trade-off between improved control performance and increased privacy risks due to occupancy sensing. More specifically, we consider occupancy-based HVAC control as the control objective and the location traces of individual occupants as the private variables. Previous studies have shown that individual location information can be inferred from occupancy measurements. To ensure privacy, we design an architecture that distorts the occupancy data in order to hide individual occupant location information while maintaining HVAC performance. Using mutual information between the individual's location trace and the reported occupancy measurement as a privacy metric, we are able to optimally design a scheme to minimize privacy risk subject to a control performance guarantee. We evaluate our framework using real-world occupancy data: first, we verify that our privacy metric accurately assesses the adversary's ability to infer private variables from the distorted sensor measurements; then, we show that control performance is maintained through simulations of building operations using these distorted occupancy readings.

cs.CR↗

arXiv2016

Dissimilarity-based Sparse Subset Selection

Finding an informative subset of a large collection of data points or models is at the center of many problems in computer vision, recommender systems, bio/health informatics as well as image and natural language processing. Given pairwise dissimilarities between the elements of a `source set' and a `target set,' we consider the problem of finding a subset of the source set, called representatives or exemplars, that can efficiently describe the target set. We formulate the problem as a row-sparsity regularized trace minimization problem. Since the proposed formulation is, in general, NP-hard, we consider a convex relaxation. The solution of our optimization finds representatives and the assignment of each element of the target set to each representative, hence, obtaining a clustering. We analyze the solution of our proposed optimization as a function of the regularization parameter. We show that when the two sets jointly partition into multiple groups, our algorithm finds representatives from all groups and reveals clustering of the sets. In addition, we show that the proposed framework can effectively deal with outliers. Our algorithm works with arbitrary dissimilarities, which can be asymmetric or violate the triangle inequality. To efficiently implement our algorithm, we consider an Alternating Direction Method of Multipliers (ADMM) framework, which results in quadratic complexity in the problem size. We show that the ADMM implementation allows to parallelize the algorithm, hence further reducing the computational time. Finally, by experiments on real-world datasets, we show that our proposed algorithm improves the state of the art on the two problems of scene categorization using representative images and time-series modeling and segmentation using representative~models.

cs.LG↗

arXiv2016

Diagnosis and Repair for Synthesis from Signal Temporal Logic Specifications

We address the problem of diagnosing and repairing specifications for hybrid systems formalized in signal temporal logic (STL). Our focus is on the setting of automatic synthesis of controllers in a model predictive control (MPC) framework. We build on recent approaches that reduce the controller synthesis problem to solving one or more mixed integer linear programs (MILPs), where infeasibility of a MILP usually indicates unrealizability of the controller synthesis problem. Given an infeasible STL synthesis problem, we present algorithms that provide feedback on the reasons for unrealizability, and suggestions for making it realizable. Our algorithms are sound and complete, i.e., they provide a correct diagnosis, and always terminate with a non-trivial specification that is feasible using the chosen synthesis method, when such a solution exists. We demonstrate the effectiveness of our approach on the synthesis of controllers for various cyber-physical systems, including an autonomous driving application and an aircraft electric power system.

eess.SY↗

arXiv2016

Differential Privacy of Populations in Routing Games

As our ground transportation infrastructure modernizes, the large amount of data being measured, transmitted, and stored motivates an analysis of the privacy aspect of these emerging cyber-physical technologies. In this paper, we consider privacy in the routing game, where the origins and destinations of drivers are considered private. This is motivated by the fact that this spatiotemporal information can easily be used as the basis for inferences for a person's activities. More specifically, we consider the differential privacy of the mapping from the amount of flow for each origin-destination pair to the traffic flow measurements on each link of a traffic network. We use a stochastic online learning framework for the population dynamics, which is known to converge to the Nash equilibrium of the routing game. We analyze the sensitivity of this process and provide theoretical guarantees on the convergence rates as well as differential privacy values for these models. We confirm these with simulations on a small example.

cs.CR↗

arXiv2015

Quantifying the Utility-Privacy Tradeoff in the Smart Grid

The modernization of the electrical grid and the installation of smart meters come with many advantages to control and monitoring. However, in the wrong hands, the data might pose a privacy threat. In this paper, we consider the tradeoff between smart grid operations and the privacy of consumers. We analyze the tradeoff between smart grid operations and how often data is collected by considering a realistic direct-load control example using thermostatically controlled loads, and we give simulation results to show how its performance degrades as the sampling frequency decreases. Additionally, we introduce a new privacy metric, which we call inferential privacy. This privacy metric assumes a strong adversary model, and provides an upper bound on the adversary's ability to infer a private parameter, independent of the algorithm he uses. Combining these two results allow us to directly consider the tradeoff between better load control and consumer privacy.

cs.CR↗

arXiv2015

Event-Selected Vector Field Discontinuities Yield Piecewise-Differentiable Flows

We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few domains of application. Under the conditions that (i) the vector field's discontinuities are locally confined to a finite number of smooth submanifolds and (ii) the vector field is transverse to these surfaces in an appropriate sense, we show that the vector field yields a well-defined flow that is Lipschitz continuous and piecewise-differentiable. This implies that although the flow is not classically differentiable, nevertheless it admits a first-order approximation (known as a Bouligand derivative) that is piecewise-linear and continuous at every point. We exploit this first-order approximation to infer existence of piecewise-differentiable impact maps (including Poincaré maps for periodic orbits), show the flow is locally conjugate (via a piecewise-differentiable homeomorphism) to a flowbox, and assess the effect of perturbations (both infinitesimal and non-infinitesimal) on the flow. We use these results to give a sufficient condition for the exponential stability of a periodic orbit passing through a point of multiply intersecting events, and apply the theory in illustrative examples to demonstrate synchronization in abstract first- and second-order phase oscillator models.

math.DS↗