Contingency Games for Multi-Agent Interaction

Contingency planning, wherein an agent generates a set of possible plans conditioned on the outcome of an uncertain event, is an increasingly popular way for robots to act under uncertainty. In this work, we take a game-theoretic perspective on contingency planning which is tailored to multi-agent scenarios in which a robot's actions impact the decisions of other agents and vice versa. The resulting contingency game allows the robot to efficiently coordinate with other agents by generating strategic motion plans conditioned on multiple possible intents for other actors in the scene. Contingency games are parameterized via a scalar variable which represents a future time at which intent uncertainty will be resolved. Varying this parameter enables a designer to easily adjust how conservatively the robot behaves in the game. Interestingly, we also find that existing variants of game-theoretic planning under uncertainty are readily obtained as special cases of contingency games. Lastly, we offer an efficient method for solving N-player contingency games with nonlinear dynamics and non-convex costs and constraints. Through a series of simulated autonomous driving scenarios, we demonstrate that plans generated via contingency games provide quantitative performance gains over game-theoretic motion plans that do not account for future uncertainty reduction.

Cost Inference for Feedback Dynamic Games from Noisy Partial State Observations and Incomplete Trajectories

In multi-agent dynamic games, the Nash equilibrium state trajectory of each agent is determined by its cost function and the information pattern of the game. However, the cost and trajectory of each agent may be unavailable to the other agents. Prior work on using partial observations to infer the costs in dynamic games assumes an open-loop information pattern. In this work, we demonstrate that the feedback Nash equilibrium concept is more expressive and encodes more complex behavior. It is desirable to develop specific tools for inferring players' objectives in feedback games. Therefore, we consider the dynamic game cost inference problem under the feedback information pattern, using only partial state observations and incomplete trajectory data. To this end, we first propose an inverse feedback game loss function, whose minimizer yields a feedback Nash equilibrium state trajectory closest to the observation data. We characterize the landscape and differentiability of the loss function. Given the difficulty of obtaining the exact gradient, our main contribution is an efficient gradient approximator, which enables a novel inverse feedback game solver that minimizes the loss using first-order optimization. In thorough empirical evaluations, we demonstrate that our algorithm converges reliably and has better robustness and generalization performance than the open-loop baseline method when the observation data reflects a group of players acting in a feedback Nash game.

Towards Dynamic Causal Discovery with Rare Events: A Nonparametric Conditional Independence Test

Causal phenomena associated with rare events occur across a wide range of engineering problems, such as risk-sensitive safety analysis, accident analysis and prevention, and extreme value theory. However, current methods for causal discovery are often unable to uncover causal links, between random variables in a dynamic setting, that manifest only when the variables first experience low-probability realizations. To address this issue, we introduce a novel statistical independence test on data collected from time-invariant dynamical systems in which rare but consequential events occur. In particular, we exploit the time-invariance of the underlying data to construct a superimposed dataset of the system state before rare events happen at different timesteps. We then design a conditional independence test on the reorganized data. We provide non-asymptotic sample complexity bounds for the consistency of our method, and validate its performance across various simulated and real-world datasets, including incident data collected from the Caltrans Performance Measurement System (PeMS). Code containing the datasets and experiments is publicly available.

SLAM Backends with Objects in Motion: A Unifying Framework and Tutorial

Simultaneous Localization and Mapping (SLAM) algorithms are frequently deployed to support a wide range of robotics applications, such as autonomous navigation in unknown environments, and scene mapping in virtual reality. Many of these applications require autonomous agents to perform SLAM in highly dynamic scenes. To this end, this tutorial extends a recently introduced, unifying optimization-based SLAM backend framework to environments with moving objects and features. Using this framework, we consider a rapprochement of recent advances in dynamic SLAM. Moreover, we present dynamic EKF SLAM: a novel, filtering-based dynamic SLAM algorithm generated from our framework, and prove that it is mathematically equivalent to a direct extension of the classical EKF SLAM algorithm to the dynamic environment setting. Empirical results with simulated data indicate that dynamic EKF SLAM can achieve high localization and mobile object pose estimation accuracy, as well as high map precision, with high efficiency.

Pursuit of a Discriminative Representation for Multiple Subspaces via Sequential Games

We consider the problem of learning discriminative representations for data in a high-dimensional space with distribution supported on or around multiple low-dimensional linear subspaces. That is, we wish to compute a linear injective map of the data such that the features lie on multiple orthogonal subspaces. Instead of treating this learning problem using multiple PCAs, we cast it as a sequential game using the closed-loop transcription (CTRL) framework recently proposed for learning discriminative and generative representations for general low-dimensional submanifolds. We prove that the equilibrium solutions to the game indeed give correct representations. Our approach unifies classical methods of learning subspaces with modern deep learning practice, by showing that subspace learning problems may be provably solved using the modern toolkit of representation learning. In addition, our work provides the first theoretical justification for the CTRL framework, in the important case of linear subspaces. We support our theoretical findings with compelling empirical evidence. We also generalize the sequential game formulation to more general representation learning problems. Our code, including methods for easy reproduction of experimental results, is publically available on GitHub.

GTP-SLAM: Game-Theoretic Priors for Simultaneous Localization and Mapping in Multi-Agent Scenarios

Robots operating in complex, multi-player settings must simultaneously model the environment and the behavior of human or robotic agents who share that environment. Environmental modeling is often approached using Simultaneous Localization and Mapping (SLAM) techniques; however, SLAM algorithms usually neglect multi-player interactions. In contrast, a recent branch of the motion planning literature uses dynamic game theory to explicitly model noncooperative interactions of multiple agents in a known environment with perfect localization. In this work, we fuse ideas from these disparate communities to solve SLAM problems with game theoretic priors. We present GTP-SLAM, a novel, iterative best response-based SLAM algorithm that accurately performs state localization and map reconstruction in an uncharted scene, while capturing the inherent game-theoretic interactions among multiple agents in that scene. By formulating the underlying SLAM problem as a potential game, we inherit a strong convergence guarantee. Empirical results indicate that, when deployed in a realistic traffic simulation, our approach performs localization and mapping more accurately than a standard bundle adjustment algorithm across a wide range of noise levels.

Zeroth-Order Methods for Convex-Concave Minmax Problems: Applications to Decision-Dependent Risk Minimization

Min-max optimization is emerging as a key framework for analyzing problems of robustness to strategically and adversarially generated data. We propose a random reshuffling-based gradient free Optimistic Gradient Descent-Ascent algorithm for solving convex-concave min-max problems with finite sum structure. We prove that the algorithm enjoys the same convergence rate as that of zeroth-order algorithms for convex minimization problems. We further specialize the algorithm to solve distributionally robust, decision-dependent learning problems, where gradient information is not readily available. Through illustrative simulations, we observe that our proposed approach learns models that are simultaneously robust against adversarial distribution shifts and strategic decisions from the data sources, and outperforms existing methods from the strategic classification literature.

The Computation of Approximate Generalized Feedback Nash Equilibria

We present the concept of a Generalized Feedback Nash Equilibrium (GFNE) in dynamic games, extending the Feedback Nash Equilibrium concept to games in which players are subject to state and input constraints. We formalize necessary and sufficient conditions for (local) GFNE solutions at the trajectory level, which enable the development of efficient numerical methods for their computation. Specifically, we propose a Newton-style method for finding game trajectories which satisfy the necessary conditions, which can then be checked against the sufficiency conditions. We show that the evaluation of the necessary conditions in general requires computing a series of nested, implicitly-defined derivatives, which quickly becomes intractable. To this end, we introduce an approximation to the necessary conditions which is amenable to efficient evaluation, and in turn, computation of solutions. We term the solutions to the approximate necessary conditions Generalized Feedback Quasi Nash Equilibria (GFQNE), and we introduce numerical methods for their computation. In particular, we develop a Sequential Linear-Quadratic Game approach, in which a locally approximate LQ game is solved at each iteration. The development of this method relies on the ability to compute a GFNE to inequality- and equality-constrained LQ games, and therefore specific methods for the solution of these special cases are developed in detail. We demonstrate the effectiveness of the proposed solution approach on a dynamic game arising in an autonomous driving application.

Multi-Hypothesis Interactions in Game-Theoretic Motion Planning

We present a novel method for handling uncertainty about the intentions of non-ego players in dynamic games, with application to motion planning for autonomous vehicles. Equilibria in these games explicitly account for interaction among other agents in the environment, such as drivers and pedestrians. Our method models the uncertainty about the intention of other agents by constructing multiple hypotheses about the objectives and constraints of other agents in the scene. For each candidate hypothesis, we associate a Bernoulli random variable representing the probability of that hypothesis, which may or may not be independent of the probability of other hypotheses. We leverage constraint asymmetries and feedback information patterns to incorporate the probabilities of hypotheses in a natural way. Specifically, increasing the probability associated with a given hypothesis from $0$ to $1$ shifts the responsibility of collision avoidance from the hypothesized agent to the ego agent. This method allows the generation of interactive trajectories for the ego agent, where the level of assertiveness or caution that the ego exhibits is directly related to the easy-to-model uncertainty it maintains about the scene.

Encoding Defensive Driving as a Dynamic Nash Game

Robots deployed in real-world environments should operate safely in a robust manner. In scenarios where an "ego" agent navigates in an environment with multiple other "non-ego" agents, two modes of safety are commonly proposed: adversarial robustness and probabilistic constraint satisfaction. However, while the former is generally computationally-intractable and leads to overconservative solutions, the latter typically relies on strong distributional assumptions and ignores strategic coupling between agents. To avoid these drawbacks, we present a novel formulation of robustness within the framework of general sum dynamic game theory, modeled on defensive driving. More precisely, we inject the ego's cost function with an adversarial phase, a time interval during which other agents are assumed to be temporarily distracted, to robustify the ego agent's trajectory against other agents' potentially dangerous behavior during this time. We demonstrate the effectiveness of our new formulation in encoding safety via multiple traffic scenarios.